Key Concept: Combining Arithmetic Expressions and Order of Operations

a) $(8 + 4 \times 2)$ is greater
[Solution Description]
To find out which expression is greater, we need to evaluate both expressions by following the order of operations.
For the expression $(8 + 4 \times 2)$:
According to the order of operations, multiplication comes before addition. So calculate $4 \times 2$ first:
$4 \times 2 = 8$
Then add 8 to the result:
$8 + 8 = 16$
For the expression $(12 \div 3 + 9)$:
According to the order of operations, division comes before addition. So calculate $12 \div 3$ first:
$12 \div 3 = 4$
Then add 9 to the result:
$4 + 9 = 13$
Comparing the two results, $16 > 13$, so the expression $(8 + 4 \times 2)$ is greater.

Key Concept: Expression Equivalent and Distributive Property

a) $12 + y$
[Solution Description]
First, apply the distributive property to expand $3(4 + y)$:
Multiply 3 with each term inside the parentheses:
$3 \times 4 + 3 \times y = 12 + 3y$
Substitute this expanded form back into the original expression:
$12 + 3y – 2y$
Now combine like terms by subtracting $2y$ from $3y$:
$12 + (3y – 2y) = 12 + y$
Thus, the expression is equivalent to $12 + y$.

Key Concept: Comparing Expressions

b) False
[Solution Description]
We will evaluate each side of the inequality separately:
Left side: $12 + 7 = 19$
Right side: $9 + 11 = 20$
Now compare the two values:
Since $19 9 + 11$ is false.

Key Concept: Definition of Arithmetic Expressions

b) 15
[Solution Description]
The arithmetic expression given is $7 + 8$. In order to find its value, we simply add 7 and 8 together:
$7 + 8 = 15$
Therefore, the value of the expression is 15.

Key Concept: Subtraction in Arithmetic Expressions

c) 15
[Solution Description]
The situation can be expressed as the arithmetic expression $20 – 5$.
By subtracting 5 from 20, we get:
$20 – 5 = 15$
Thus, John has 15 candies remaining.

Key Concept: Simple Expressions: Definition of arithmetic expressions

d) Assertion is false, but Reason is true.
[Solution Description]
To evaluate the arithmetic expression $7 + 3 \times 2$, we follow the order of operations, which dictates that multiplication is performed before addition.
First, calculate the multiplication part:
$3 \times 2 = 6$
Then add this result to 7:
$7 + 6 = 13$
Now, evaluate the expression $13 + 1$:
$13 + 1 = 14$
Since the calculation shows that $7 + 3 \times 2 = 13$ and $13 + 1 = 14$, they do not have the same value.
The assertion is false since $7 + 3 \times 2 \neq 13 + 1$. As for the reason, it states that arithmetic expressions can only be evaluated using addition and subtraction. This is incorrect because arithmetic expressions can also involve operations like multiplication and division.
Hence, both the Assertion and Reason are false.

Key Concept: Comparing Expressions

c) $15 + 5 = 10 \times 2$
[Solution Description] To compare $15 + 5$ and $10 \times 2$, calculate their values:
Expression 1: $15 + 5$
$15 + 5 = 20$
Expression 2: $10 \times 2$
$10 \times 2 = 20$
Since both expressions result in the same value, they are equal.

Key Concept: Comparison of expressions

d) $9 \times 5 > 46 – 2$
[Solution Description]
First, evaluate each expression.
For $9 \times 5$:
Calculate the product: $9 \times 5 = 45$.
For $46 – 2$:
Subtract to find the difference: $46 – 2 = 44$.
Comparing the results, $45 > 44$, so $9 \times 5 > 46 – 2$.

Key Concept: Simple Expressions

c) $6 \times 3$
[Solution Description] We need to find which expression equals $18$. Let’s evaluate each option:
Option a): $9 + 9$
$9 + 9 = 18$
Option b): $20 – 3$
$20 – 3 = 17$
Option c): $6 \times 3$
$6 \times 3 = 18$
Option d): $36 \div 2$
$36 \div 2 = 18$
Options a), c), and d) all equal $18$ but since we can choose only one, options c) and d) are more straightforward calculations with no intermediate addition or subtraction needed.

Key Concept: Comparing Expressions

b) 20
[Solution Description]
First, calculate $9 \times 4$:
$9 \times 4 = 36$.
Then, find the missing value in the equation $36 = \_\_\_ + 16$ by subtracting $16$ from $36$:
$36 – 16 = 20$.
So, the equation becomes $9 \times 4 = 20 + 16$.

Key Concept: Comparing expressions using >, <, =

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if both the assertion and reason are true, we evaluate both expressions separately:
– Calculate $150 + 35$:
$150 + 35 = 185$
– Given that the second expression is already $185$, they indeed have the same value.
Therefore, both the Assertion and the Reason are true, and the Reason correctly explains why the Assertion is true.

Key Concept: Evaluating and Comparing Nested Expressions

a) $(52 + 205) – (150 – 50) > 3 \times (100 – 90) + 20$
[Solution Description]
The first expression is evaluated as follows:
Calculate inside the innermost parentheses first:
$(150 – 50) = 100$
Then substitute back into the expression:
$(52 + 205) – 100 = 257 – 100 = 157$
Evaluating the second expression:
Inside the parentheses, calculate:
$(100 – 90) = 10$
Substitute back into the main expression:
$3 \times 10 + 20 = 30 + 20 = 50$
Now compare both results:
$157 > 50$
Hence, the first expression is greater.

Key Concept: Different expressions for same value

a) $50 – 14$
[Solution Description]
The expression $48 – 12$ can be simplified as follows:
$48 – 12 = 36$
Let’s evaluate each option:
a) $50 – 14$: Simplifies to $50 – 14 = 36$
b) $60 – 24$: Simplifies to $60 – 24 = 36$
c) $72 – 34$: Simplifies to $72 – 34 = 38$, not equal to 36.
d) $39 + (-3)$: Simplifies to $39 – 3 = 36$
Therefore, options a), b), and d) are all equivalent to $48 – 12$.
Option c) is incorrect because it does not simplify to 36.

Key Concept: Different expressions for same value

a) $45 \div 5$
[Solution Description]
The expression $81 \div 9$ results in:
$81 \div 9 = 9$
Assessing each option:
a) $45 \div 5$: This simplifies to $45 \div 5 = 9$.
b) $90 \div 10$: This simplifies to $90 \div 10 = 9$.
c) $27 \div 3$: This simplifies to $27 \div 3 = 9$.
d) $35 \div 4$: This simplifies to $35 \div 4 = 8.75$, which is not 9.
Thus, options a), b), and c) provide the same result as $81 \div 9$. Option d) does not.

Key Concept: Simple Expressions: Different expressions for same value

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the expressions $15 – 3$ and $9 + 3$ have the same value. Let’s evaluate both:
For $15 – 3$, we calculate:
$15 – 3 = 12$
For $9 + 3$, we calculate:
$9 + 3 = 12$
Both expressions indeed have the same value of 12, making the assertion true.
The reason says subtraction is not associative like addition. This means, unlike addition where $(a + b) + c = a + (b + c)$, subtraction does not hold this property. For example, $(8 – 3) – 2 = 5 – 2 = 3$ but $8 – (3 – 2) = 8 – 1 = 7$. Hence, the reason is also true.
However, the reason regarding associativity does not directly explain why the two different expressions have the same value, which is due to their individual evaluated results being equal.

Key Concept: Simple Expressions: Different expressions for same value

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this, evaluate both expressions:
For $2 \times 6$, the result is:
$2 \times 6 = 12$
For $3 + 9$, perform the addition:
$3 + 9 = 12$
Since both expressions evaluate to 12, they are indeed equivalent. Thus, the assertion that $2 \times 6$ is equivalent to $3 + 9$ is true. Furthermore, the reason provided explains adequately why the assertion is true as it states the results of both expressions being equal to 12.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Comparing expressions using $$ signs

c) $10 + 20 > 15 + 12$
[Solution Description] To solve this, we need to evaluate each side of the inequality:
1. Evaluate $10 + 20 = 30$.
2. Evaluate $15 + 12 = 27$.
Now compare $30$ and $27$:
Since $30 > 27$, the correct inequality is: $10 + 20 > 15 + 12$.

Key Concept: Simple Expressions: Ordering expressions based on value

d) Assertion is false, but Reason is true.
[Solution Description]
To evaluate whether the assertion is true, let’s first calculate the values of each expression:
– For the expression $364 + 587$, we perform the addition:
$364 + 587 = 951$
– For the expression $363 + 589$, we perform the following addition:
$363 + 589 = 952$
By comparing these calculations, $364 + 587$ equals 951 and $363 + 589$ equals 952, which are not equal.
Therefore, the Assertion is false as $364 + 587 \neq 363 + 589$.
Now, looking at the Reason: The associative property states that how numbers are grouped does not affect their sum (e.g., $(a+b)+c = a+(b+c)$). This implies rearrangement can lead to equivalent expressions when applied correctly. However, this property doesn’t support the given Assertion because the sums calculated are different.
Hence, the Reason is true since it accurately describes the property without contradicting itself.
Thus, the correct option is d).

Key Concept: Understanding and Comparing Expressions, Commutative Property

b) $273 – 145$ and $272 – 144$
[Solution Description]
Consider the two pairs:
– Pair 1: $245 + 289$ and $246 + 285$
– Compute both: $245 + 289 = 534$, $246 + 285 = 531$
– Pair 2: $273 – 145$ and $272 – 144$
– Compute both: $273 – 145 = 128$, $272 – 144 = 128$
The second pair results in equal values, so they are equal.

Key Concept: Understanding arithmetic expressions and their values

a) $24 + 36$
[Solution Description]
To determine which expression has the highest value, we calculate each:
For $24 + 36$,
$24 + 36 = 60$
For $35 + 25$,
$35 + 25 = 60$
For $40 + 15$,
$40 + 15 = 55$
For $45 + 10$,
$45 + 10 = 55$
Both $24 + 36$ and $35 + 25$ have the highest value of 60.
– Option a) $24 + 36 = 60$
– Option b) $35 + 25 = 60$
– Option c) $40 + 15 = 55$
– Option d) $45 + 10 = 55$
Since both option a) and b) are correct, without further distinction in options, either choice could be interpreted as correct depending on the context provided. However, strictly based on provided MCQ structure where one clear answer is expected, only considering first instance:
Correct Answer: a) $24 + 36$

Key Concept: Brackets in Expressions

c) 14
[Solution Description] According to the order of operations, we first perform multiplication and then addition. So, we calculate $3 \times 2 = 6$. Then add this result to 8:
$8 + 6 = 14$
Hence, the value of the expression is 14.

Key Concept: Reading and Evaluating Complex Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To correctly evaluate the expression, we need to consider the rules of operation in mathematics which suggest performing addition and subtraction from left to right without brackets. Thus, initially calculating $100 – 15 = 85$, and then adding 56 results in $85 + 56 = 141$. However, with proper interpretation using brackets, the calculation should occur as follows:
Evaluate $15 + 56 = 71$ first since this is within the bracket:
$100 – 71 = 29$
Hence, evaluating without brackets gives 141, but proper inclusion of brackets as $100 – (15 + 56)$ gives a correct result of 29.

Key Concept: Use of Brackets

b) 28
[Solution Description] Begin with the operation inside the brackets:
$9 – 2 = 7$
Then multiply this result by 3:
$7 \times 3 = 21$
Finally, add this product to 7:
$7 + 21 = 28$
Therefore, the simplified value of the expression is 28.

Key Concept: Complex Expressions, Understanding Ambiguity

d) 45
[Solution Description]
For the expression $8 \times 5 + 10 / 2$, we use the order of operations (PEMDAS/BODMAS). There are no parentheses or exponents, so we proceed with multiplication and division before addition.
Step 1: Calculate the multiplication:
$8 \times 5 = 40$
Step 2: Calculate the division:
$10 / 2 = 5$
Step 3: Substitute these results back into the expression:
$40 + 5$
Step 4: Perform the addition:
$40 + 5 = 45$
Therefore, the correct value of the expression is 45.

Key Concept: Reading and Evaluating Complex Expressions, Ambiguity without proper rules

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To evaluate the expression $500 – (250 – 100)$ correctly, we need to follow the order of operations and understand the impact of removing brackets with a preceding negative sign.
Step 1: Begin with the inner bracket subtraction:
$250 – 100 = 150$
Step 2: Substitute this value back into the main expression:
$500 – (250 – 100) = 500 – 150$
Step 3: Perform the subtraction:
$500 – 150 = 350$
The assertion that the expression evaluates to $350$ is true.
Now consider the reason: When we remove brackets preceded by a negative sign, the operation effectively turns into subtraction because all terms inside the bracket are subtracted from what comes before the bracket in the original expression. Here, $(250 – 100)$ becomes $150$, and since it was already properly processed inside the brackets before being subtracted, there was no need to change any signs upon removal of the brackets.
Thus, both the assertion and reason are correct, but the reason does not explain why the evaluation was done correctly; it’s simply an observation about changing signs upon direct removal of brackets with preceding negatives which wasn’t directly applied here.

Key Concept: Ambiguity without proper rules

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the assertion, we consider the expression $30 + 5 \times 4$. Without knowing the correct order of operations, one might mistakenly add 30 and 5 first to get 35, then multiply by 4 to get 140. This yields a result of 140 which is incorrect.
However, according to the standard mathematical rules of precedence, multiplication takes priority over addition. Thus, we first compute $5 \times 4 = 20$, and then add this product to 30:
$30 + 20 = 50$
Therefore, the assertion that the expression can be interpreted differently without a clear rule is true.
Evaluating the reason, the statement claims that multiplication should be done before addition as per mathematical conventions. This aligns with the standard order of operations (BODMAS/BIDMAS), where B stands for brackets, O/I for orders/indices (powers and roots, etc.), DM for division and multiplication (from left to right), and AS for addition and subtraction (from left to right).
Hence, the reason is also true and it correctly explains the assertion because following these rules removes ambiguity in interpreting expressions.

Key Concept: Understanding ambiguity in evaluating expressions

b) $3 + 4 \times 2$
[Solution Description]
An expression like $3 + 4 \times 2$ can be ambiguous if there is no rule specifying whether addition or multiplication should be performed first. According to standard order of operations (PEMDAS/BODMAS), multiplication should be done before addition.
Calculating based on these rules:
– First, perform the multiplication: $4 \times 2 = 8$
– Then add: $3 + 8 = 11$
If the expression is interpreted incorrectly as $(3 + 4) \times 2$, it results in a different value:
– Add first: $3 + 4 = 7$
– Then multiply: $7 \times 2 = 14$
Hence, the expression $3 + 4 \times 2$ shows how ambiguity arises without proper evaluation rules.

Key Concept: Evaluating expressions with brackets

b) 15
[Solution Description]
To resolve the expression $20 – (10 – 5)$, follow these steps:
– First, solve inside the brackets: $10 – 5 = 5$
– Substitute back into the expression: $20 – 5$
– Finally, subtract: $20 – 5 = 15$
The use of brackets clarifies that the subtraction within them must be performed first, resulting in a final value of 15.

Key Concept: Order of operations and evaluating expressions with brackets

c) Rs.1930
[Solution Description]
The initial cost of supplies is Rs.2000. The discount applied is Rs.250, so the effective cost becomes:
$2000 – 250 = 1750$
After applying the discount, a tax of Rs.180 is added:
$1750 + 180 = 1930$
Thus, the final amount paid by the company is Rs.1930.

Key Concept: Real-world Context: Expression for Total Cost

a) Rs.2500
[Solution Description]
First, calculate the total cost for adult tickets:
$8 \times 250 = 2000$
Next, calculate the total cost for child tickets:
$6 \times 150 = 900$
Add both amounts to find the gross total before the discount:
$2000 + 900 = 2900$
Apply the discount to this total:
$2900 – 400 = 2500$
Therefore, the net amount they paid is Rs.2500.

Key Concept: Complex expressions involving addition, multiplication, and division in real-world scenarios

c) Rs.1485
[Solution Description]
First, calculate the total cost without the discount:
– Books: $5 \times 300 = 1500$
– Stationery set: Rs.150
Sum up the costs:
$1500 + 150 = 1650$
Apply a 10$\%$ discount on Rs.1650:
$\text{Discount} = \frac{10}{100} \times 1650 = 165$
Subtract the discount from the total cost:
$1650 – 165 = 1485$
Therefore, the shopper pays Rs.1485.

Key Concept: Reading and Evaluating Complex Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
In the given assertion, we have an expression $100 – (15 + 56)$ which is used to determine how much change Irfan will receive after spending money on biscuits and toor dal. To evaluate this expression:
Step 1: Evaluate the expression inside the bracket first. Calculate $15 + 56$.
$15 + 56 = 71$
Step 2: Subtract the sum obtained from the amount given to the shopkeeper.
$100 – 71 = 29$
Therefore, the correct change Irfan gets back is Rs.29. This confirms that the assertion is true.
The reason provided explains the use of brackets in mathematical expressions, highlighting that operations inside the brackets should be done first as per the standard order of operations (PEMDAS/BODMAS). Both the assertion and the reason are true statements, but the reason directly supports the logic behind why the expression is structured with brackets. Hence, it is the correct explanation for evaluating the expression accurately.

Key Concept: Role of brackets in expressions with multiple operations

c) 48
[Solution Description]
Start by solving the expressions within each bracket.
For the first: $8 + 4 = 12$.
For the second: $6 – 2 = 4$.
Substitute these values into the expression: $12 \times 4$.
Multiply to find the final result: $12 \times 4 = 48$.

Key Concept: Brackets involving subtraction and addition

b) 50
[Solution Description]
First, solve the expression within the brackets: $15 – 5 = 10$.
The expression becomes $50 – 10 + 10$.
Perform the first subtraction: $50 – 10 = 40$.
Then add the remaining term: $40 + 10 = 50$.

Key Concept: Reading and Evaluating Complex Expressions, Importance of Brackets in Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion:
The original expression is $50 – (30 + 20)$. Perform operations within the brackets first:
$30 + 20 = 50$
Then subtract from 50:
$50 – 50 = 0$
Now consider removing brackets with changing signs:
Rewriting the expression as $50 – 30 – 20$:
Subtract step-by-step:
$50 – 30 = 20$
$20 – 20 = 0$
Both methods yield a result of 0, thus confirming the assertion that removing the brackets yields an equivalent expression.
To verify the reason:
According to arithmetic rules, when brackets are preceded by a negative sign, it indeed affects the signs of the enclosed terms. For example, converting $-(a+b)$ into $-a-b$.
Therefore, both the Assertion and Reason are true, and the Reason correctly explains why removing the brackets in this manner yields an equivalent result.

Key Concept: Importance of brackets in expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the expression $100 – (15 + 56)$ simplifies to $100 – 71$. This is accurate because according to the order of operations, calculations within the brackets are performed first. Evaluating $(15 + 56)$ gives $71$, and thus the whole expression becomes $100 – 71 = 29$.
The reason supports this by explaining the role of brackets in guiding the evaluation order. It correctly states that operations within brackets take precedence.
Therefore, both the assertion and reason are true, and the reason is the correct explanation for the assertion.

Key Concept: Evaluating with and without brackets, Understanding the commutative and associative properties

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To evaluate whether the assertion and reason are true, we look at the given expression $125 + 36 – 20$ and compare it with $(125 + 36) – 20$.
Let’s solve each:
Without brackets: $125 + 36 – 20$
$= 161 – 20 = 141$
With brackets: $(125 + 36) – 20$
$= 161 – 20 = 141$
Both expressions give the same result, which confirms that they are equivalent.
The reason states that brackets indicate priority in operations. In this specific case, adding brackets did not change the outcome because the operations are executed from left to right; however, in more complex expressions, brackets can indeed affect the result by altering the order of operations.
The assertion is true: the two expressions are equivalent.
The reason is also true but doesn’t explain why these two particular forms yield the same results since they naturally follow left-to-right execution in simple arithmetic without division or multiplication influencing priorities.
Thus, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Removing brackets with a negative sign

b) $45$
[Solution Description]
Start by calculating the sum inside the brackets:
$30 + 25 = 55$
Then, apply the subtraction outside the brackets:
$100 – 55 = 45$
Therefore, the evaluated value of the expression is $45$.

Key Concept: Evaluating expressions with brackets

c) $29$
[Solution Description] Solving this requires evaluating inside the brackets first. Begin with adding the numbers within the brackets:
$15 + 56 = 71$
Next, subtract from 100:
$100 – 71 = 29$
Thus, $100 – (15 + 56) = 29$.

Key Concept: Using distributive property

d) $64$
[Solution Description]
First, simplify within the parentheses:
$4 + 6 = 10$
Multiply this result by $7$:
$10 \times 7 = 70$
Calculate $3 \times 2$ as follows:
$3 \times 2 = 6$
Subtract the second result from the first:
$70 – 6 = 64$
Thus, the simplified expression evaluates to $64$.

Key Concept: Reading and Evaluating Complex Expressions: Brackets resolve ambiguity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this problem, let’s evaluate the assertion (A) by computing the expression $100 – (15 + 56)$ with and without considering brackets:
Without using brackets:
$100 – 15 + 56 = 85 + 56 = 141$
With brackets properly used as in $100 – (15 + 56)$:
First, calculate inside the brackets:
$15 + 56 = 71$
Then subtract from 100:
$100 – 71 = 29$
The use of brackets ensures that the addition is completed before subtraction, giving us the correct result. Thus, Assertion (A) is true.
Additionally, Reason (R) states that brackets specify the order of operations and help avoid ambiguities in calculations. This statement holds true as demonstrated. Evaluating what’s inside the brackets first clarifies the operation sequence and results in a different outcome compared to if brackets were ignored.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Arithmetic Expressions

c) 80
[Solution Description]
To evaluate the expression, we first need to resolve the brackets:
Inside the brackets: $8 – 3 = 5$.
Now substitute back into the original expression: $50 + 6 \times 5$.
Next, perform the multiplication: $6 \times 5 = 30$.
Finally, add this result to $50$: $50 + 30 = 80$.
Therefore, the value of the expression is $80$.

Key Concept: Reading and Evaluating Complex Expressions, Brackets resolve ambiguity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To evaluate the expression $100 – 50 + 20$, we must understand the order of operations. By default, addition and subtraction are performed from left to right. Therefore,
Without brackets:
$100 – 50 + 20 = (100 – 50) + 20 = 50 + 20 = 70$
With appropriate brackets such as $(100 – 50) + 20$, the result remains 70, confirming that brackets can help ensure clarity but do not change the evaluation in this specific instance.
The assertion is true because the expression evaluates to 70 with or without brackets when following standard operator precedence rules (left-to-right for addition and subtraction). However, the reason given is true universally, stating a general principle about how brackets provide clarity in determining operation sequences. Thus, both statements are true, and the reason is the correct explanation for why using brackets is beneficial.

Key Concept: Resolving Ambiguity with Brackets, Order of Operations

d) 48
[Solution Description]
The given expression is $80 – [4 \times (6 + 2)]$. To resolve the ambiguity and correctly evaluate it, follow the order of operations:
Step 1: Solve inside the innermost brackets first: $(6 + 2) = 8$.
Step 2: Multiply within the square brackets: $4 \times 8 = 32$.
Step 3: Subtract from 80: $80 – 32 = 48$.
Therefore, the correct evaluation of the expression is 48.

Key Concept: Conversion of Subtraction in Expressions

b) $5 + (-8) + 12$
[Solution Description]
The expression $5 – 8 + 6 \times 2$ needs subtraction converted to addition:
Step 1: Convert subtraction to addition:
$5 – 8 + 6 \times 2 = 5 + (-8) + 6 \times 2$
Here, $5$, $-8$, and $6 \times 2$ are the separate terms.
Therefore, the expression as the sum of its terms is $5 + (-8) + 12$, since $6 \times 2 = 12$.

Key Concept: Terms in Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that in the expression $9 + 4 – 7$, the terms are $9$, $4$, and $-7$. According to the definition of terms, these are parts of an expression separated by a ‘+’ sign or can be perceived as such when subtraction is rewritten as addition of negatives. Thus, this assertion is true.
The reason provides a valid explanation stating that subtracting a number is equivalent to adding its inverse. Hence, negative numbers like $-7$ can indeed be considered as separate terms within expressions.
Therefore, both the Assertion and Reason are true, and the Reason acts as the correct explanation for the Assertion.

Key Concept: Order of operations, Conversion of subtraction to addition

d) 45
[Solution Description]
First, identify the terms in the expression: $18 \times 3$, $-12$, and $6 \div 2$.
Convert the subtraction to addition by adding the negative: $18 \times 3 + (-12) + 6 \div 2$.
Evaluate multiplication and division first:
Calculate $18 \times 3 = 54$.
Calculate $6 \div 2 = 3$.
Now the expression becomes $54 + (-12) + 3$.
Add these values sequentially:
Add $54$ and $(-12)$ which gives $42$.
Then add $42$ and $3$ which gives $45$.
So, the result is $45$.

Key Concept: Understanding the commutative property

a) 13 + 7
[Solution Description]
The commutative property states that numbers can be added in any order without changing the result. Therefore, $7 + 13$ is equivalent to $13 + 7$ when applying the commutative property of addition.

Key Concept: Identifying Terms

a) 45, 7, -3
[Solution Description] To find the terms in the expression, we need to look for parts separated by plus signs. The expression is given as $45 + 7 – 3$. Subtraction can be converted to addition of negative numbers, so it becomes $45 + 7 + (-3)$. Thus, the terms are 45, 7, and -3.

Key Concept: Order of Operations, Meaning of Terms

c) 3
[Solution Description]
To identify the number of terms in the expression $7 \times 3 – 5 + 9$, we first need to rewrite any subtractions as additions of their inverse.
The expression $7 \times 3 – 5 + 9$ can be rewritten by replacing subtraction with addition:
$7 \times 3 + (-5) + 9$
Now, let’s separate the terms based on the ‘+’ sign:
– $(7 \times 3)$ is one term since multiplication doesn’t break a term.
– $(-5)$ is another term as it’s separated by a ‘+’ sign after conversion from subtraction.
– $9$ is the third term.
So, the expression has three terms: $7 \times 3$, $-5$, and $9$.

Key Concept: Inverse Concept, Order of Operations

b) $3$ terms; value = $-13$
[Solution Description]
Let’s evaluate the inner operation first and then convert the expression for clarity.
Step 1: Calculate inside the parentheses:
$(2 – 7) = -5$
Step 2: Replace and simplify:
$10 + 4 \times (-5) – 3$
Convert subtractions to additions:
$10 + 4 \times (-5) + (-3)$
Evaluate:
$10 + (-20) + (-3)$
Simplify step-by-step:
$10 + (-20) = -10$
$-10 + (-3) = -13$
Identifying terms:
– $10$ becomes a term.
– $4 \times (-5)$ is a single term due to multiplication.
– $-3$ is a term.
Total terms: Three. Value: $-13$.

Key Concept: Terms in Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] To determine terms in an expression, we first resolve multiplication and rewrite any subtraction as addition of an inverse. In $8 + 3 \times 5 – 7$, $\times$ has higher precedence than $+$ or $-$, so compute $3 \times 5 = 15$. The expression becomes $8 + 15 – 7$. Now, convert $-7$ to $+(-7)$ for clarity: $8 + 15 + (-7)$. Therefore, the terms are indeed $8$, $15$, and $-7$ which aligns with Assertion. The Reason correctly explains that rewriting helps identify terms by changing subtraction to addition of inverses, so both Assertion and Reason are true and Reason is the correct explanation of Assertion.

Key Concept: Conceptual Complexity, Distributive Property and Addition/Subtraction

Correct Answer option b) Three terms
[Solution Description]
Begin by evaluating the expression inside the brackets:
$3 \times 2 + 1 = 6 + 1 = 7$. Therefore, the expression becomes $9 – 7 + 5$.
Convert the subtraction into addition using the inverse:
$9 – 7 + 5 = 9 + (-7) + 5$
The resulting terms are $9$, $-7$, and $5$.

Key Concept: Definition of terms and their identification

d) $9$, $-3$, $15 \times 2$, $-11$
[Solution Description] Start by converting the subtractions into additions: $9 – 3$ becomes $9 + (-3)$ and $15 \times 2 – 11$ becomes $15 \times 2 + (-11)$. Hence, the expression transforms into $9 + (-3) + 15 \times 2 + (-11)$. This results in four separate terms identified by the $+$ symbol: $9$, $-3$, $15 \times 2$, and $-11$.

Key Concept: Understanding expressions without brackets

d) $45$, $-12$, and $8 \times 3$
[Solution Description] To find the terms of the expression $45 – 12 + 8 \times 3$, we need to first convert any subtraction into addition. So, $45 – 12$ can be rewritten as $45 + (-12)$. The expression then becomes $45 + (-12) + 8 \times 3$. Now the terms are those separated by $+$ signs, which gives us three terms: $45$, $-12$, and $8 \times 3$.

Key Concept: Converting subtraction to addition to identify terms

d) $10$, $5$, $-7 \times 4$, $6$
[Solution Description] In the expression $10 + 5 – 7 \times 4 + 6$, convert the subtraction into addition. Thus, $5 – 7 \times 4$ is rewritten as $5 + (-7 \times 4)$. Therefore, the entire expression can now be regarded as $10 + 5 + (-7 \times 4) + 6$. The terms separated by $+$ signs are $10$, $5$, $-7 \times 4$, and $6$.

Key Concept: Order and Grouping in Expressions

a) $12 + (-5) + (-7)$
[Solution Description] The expression $(-5) + 12 + (-7)$ involves only additions and can be rearranged due to the commutative property. Let us evaluate the original expression first:
$(-5) + 12 + (-7) = (-5) + (-7) + 12 = -12 + 12 = 0$
Now let’s verify each option:
a) $12 + (-5) + (-7) = 12 + (-12) = 0$
b) $(-5) + (-7) + 12 = (-12) + 12 = 0$
c) $(-7) + 12 + (-5) = (-7) + 7 = 0$
d) $12 + (-7) + (-5) = 12 + (-12) = 0$
All options give the result $0$, hence all rearrangements yield the same result.

Key Concept: Subtraction as Addition of Inverses

c) 20
[Solution Description]
Convert the expression $50 – (20 + 10)$ by treating subtraction as addition of inverses.
Step 1: Recognize that the negative sign before the bracket changes all signs inside the bracket:
$50 – (20 + 10) = 50 – 20 – 10$
Step 2: Perform the operations:
$= 30 – 10 = 20$
Thus, the result is 20.

Key Concept: Expression with Brackets and Sign Change

b) $33$
[Solution Description] To transform the expression $42 – (18 – 9)$ into one without brackets, we need to apply the rule that when a bracket is preceded by a minus sign, the signs inside change:
$42 – (18 – 9) = 42 – 18 + 9$
Evaluate the new expression:
$42 – 18 + 9 = 24 + 9 = 33$
So, the transformed and evaluated expression equals $33$.

Key Concept: Subtraction as addition of inverses

Correct Answer option Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] For the assertion, we have the expression $50 – 20 + 10$. Rewriting the subtraction as the addition of an inverse, it becomes $50 + (-20) + 10$. Now, calculating step-by-step:
First, add $50$ and $-20$:
$50 + (-20) = 30$
Next, add $30$ and $10$:
$30 + 10 = 40$
Thus, the expression simplifies to $40$.
For the reason, subtracting a number is indeed equivalent to adding its inverse because subtraction can be represented by adding the opposite. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Complex expression involving multiplication, division, and addition

b) $12$, $36/6 \times 3$, $-8$
[Solution Description]
Rewrite the subtraction as addition:
$12 + 36 \div 6 \times 3 – 8 = 12 + 36/6 \times 3 + (-8)$.
The expression splits at ‘+’ signs into:
– $12$
– $36/6 \times 3$
– $-8$
Multiplication and division remain combined, encapsulating this portion as a single term.

Key Concept: Conceptual Complexity, Multi-step Solutions

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
For the assertion: The expression is $10 \times 3 + 5 – 2 \div 1$. To identify the terms, we consider operations separated by a ‘+’ sign after performing all multiplications and divisions:
1. Calculate $10 \times 3 = 30$.
2. Calculate $2 \div 1 = 2$.
Now, rewrite the expression as $30 + 5 – 2$. Here, the terms separated by ‘+’ or the inverse of ‘-‘ are $30$, $5$, and $-2$.
Hence, Assertion is true because the terms mentioned in the assertion are correct.
For the reason: According to the order of operations (BODMAS/BIDMAS), multiplication and division take precedence over addition and subtraction. Thus, calculations for $10 \times 3 = 30$ and $2 \div 1 = 2$ must be executed before adding and subtracting other numbers.
Therefore, Reason is also true, but it does not directly explain how terms are identified from the expression. It explains the calculation order rather than identification of terms which is done after performing these calculations.
The correct answer is that both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Identification of terms, integration of division

d) $30/5$, $-7$, $2 \times 3$
[Solution Description]
To identify the terms in the expression, first rewrite subtraction as addition:
$30 \div 5 – 7 + 2 \times 3 = 30/5 + (-7) + 2 \times 3$.
The terms are separated by a ‘+’ sign. Therefore, the expression contains the following terms:
– $30/5$
– $-7$
– $2 \times 3$
These three components cannot be split further into individual numbers because they each represent a single term due to multiplication or division not being separated by additional ‘+’ signs.

Key Concept: Identification of terms

b) $12 \times 3$, $8 \div 4$
[Solution Description] Consider the expression $12 \times 3 + 8 \div 4$:
– $12 \times 3$ remains a single term because multiplication doesn’t split terms.
– Similarly, $8 \div 4$ is also a singular term.
The addition between these two results indicates they are separate terms. So, the expression has the terms:
– $12 \times 3$
– $8 \div 4$
Therefore, the correct answer includes both operations as individual terms.

Key Concept: Identifying Terms in Expressions

a) 45, -9, and $3 \times 6$
[Solution Description] To identify the terms in the expression $45 – 9 + 3 \times 6$, we first rewrite the subtraction as addition of the inverse: $45 + (-9) + 3 \times 6$. The terms are then separated by the ‘+’ sign. Thus, the terms are 45, -9, and $3 \times 6$.

Key Concept: Commutative Property, Associative Property

a) $(7 + (-3)) + (12 \times 5)$
[Solution Description]
We need to use the commutative and associative properties to rewrite the given expression $(12 \times 5) + (7 – 3)$.
Commutative property allows us to swap terms in addition or multiplication without changing the result; however, in this case, let’s focus on grouping them differently:
Step 1: Recognize that $7 – 3$ can be rewritten as $7 + (-3)$.
Step 2: Using the associative property, we group $12 \times 5$ separately and consider $7 + (-3)$ together, knowing from associative property of addition that grouping does not change the sum:
$(12 \times 5) + ((7 + (-3)))$
Alternatively, the expression could also be grouped as:
$12 \times (5 + 0) + (7 – 3)$
However, focusing only on alternate expressions using basic laws:
$60 + 4$
Thus, the equivalent simplified form should remain logically consistent across logical grouping, which results in:
$60 + 4 = 64$
Therefore, this expression simplifies to:
$(7 + (-3)) + (12 \times 5)$

Key Concept: Expression and Term Identification

a) $7 \times 8$, -5, and 10
[Solution Description] The expression $7 \times 8 – 5 + 10$ can be rewritten to change subtraction into addition: $7 \times 8 + (-5) + 10$. Here, $7 \times 8$, -5, and 10 are the terms since they are separated by the ‘+’ sign.

Key Concept: Conceptual Understanding through Token Model

a) 28, 4, and $-12 \times 2$
[Solution Description] Rewriting the subtraction as addition using the token model, the expression becomes $28 + 4 + (-12 \times 2)$. Therefore, the terms, separated by the ‘+’ sign, are 28, 4, and $-12 \times 2$.

Key Concept: Commutative and Associative Properties, Mixed Terms

c) The sum is unchanged at 10
[Solution Description] Using the commutative and associative properties of addition allows us to rearrange and group terms without changing the total sum. Let’s evaluate both arrangements:
Original expression:
$12 + (-4) + 9 + (-7) = (12 + (-4)) + (9 + (-7)) = 8 + 2 = 10$
Rearranged expression:
$(9 + 12) + ((-4) + (-7)) = 21 + (-11) = 10$
The sum remains 10 in both cases.
Hence, rearranging the terms doesn’t affect the sum.

Key Concept: Associative Property with Mixed Numbers

b) $4 + ((-6) + 9)$
[Solution Description] The associative property allows us to group terms in an expression differently but still get the same result. We need to check for $4 + (-6) + 9$ how we could regroup:
Originally:
$(4 + (-6)) + 9 = -2 + 9 = 7$
Rearrange using associative property:
$4 + ((-6) + 9)$
Calculate the inner brackets first:
$(-6) + 9 = 3$
Then add 4:
$4 + 3 = 7$
Both calculations give us the same result of 7.

Key Concept: Commutative Property of Addition

c) Assertion is true, but Reason is false.
[Solution Description] The commutative property of addition states that swapping the order of terms in an addition does not change the sum. For the given expressions, both calculate to $5$:
$$8 + (-3) = 5$$ and
$$(-3) + 8 = 5.$$
Hence, the Assertion is true. However, the Reason incorrectly states that swapping terms changes the value, which contradicts the commutative property. Therefore, the Reason is false.

Key Concept: Sum of Terms in Different Order

d) All of the above
[Solution Description] The commutative property states that the order of addition does not affect the outcome. Let’s evaluate $4 + 9 + (-7)$:
Simplifying: $13 – 7 = 6$
Rearranging and simplifying each option:
$4 + (-7) + 9$: $-3 + 9 = 6$
$(-7) + 4 + 9$: $-3 + 9 = 6$
$9 + 4 + (-7)$: $13 – 7 = 6$
All give the same sum of $6$.

Key Concept: Grouping Terms

a) $2 + (7 + 4)$
[Solution Description]
The concept of grouping terms allows us to group them in a different manner without changing the sum. Here, $(2 + 7) + 4$ can be grouped as $2 + (7 + 4)$.
Step 1: Compute $7 + 4$:
$7 + 4 = 11$
Step 2: Add $2$ to the result from Step 1:
$2 + 11 = 13$
Both calculations show that the value remains $13$ regardless of how the terms are grouped.

Key Concept: Associative Property of Addition

c) $11$
[Solution Description] By the associative property of addition, we can group the terms differently without changing the sum. Consider the expression $(9 + 7) + (-5)$. First, group the last two terms: $9 + (7 + (-5))$. Calculate inside the parentheses first: $7 + (-5) = 2$. Therefore, the expression becomes $9 + 2 = 11$.

Key Concept: Commutative Property, Token Model of Integers

a) $11$
[Solution Description]
Original Expression: $8 + (-6) \times 2 + 15$
Step 1: Simplify the multiplication first:
$$
(-6) \times 2 = -12
$$
So, the expression becomes:
$$
8 + (-12) + 15
$$
Step 2: Use commutative property and rearrange:
$$
(8 + 15) + (-12)
$$
Calculate $8 + 15$:
$$
8 + 15 = 23
$$
Now compute $23 + (-12)$:
$$
23 – 12 = 11
$$
Hence, the value of the expression when arranged in this manner is $11$.

Key Concept: Associative Property, Multiplication and Addition

c) $42$
[Solution Description]
Original Expression: $2 \times (3 + 4) + 18 + 5 \times (1 + 1)$
Step 1: Simplify within parentheses:
$$
3 + 4 = 7
$$
And,
$$
1 + 1 = 2
$$
Substituting these back into the expression gives:
$$
2 \times 7 + 18 + 5 \times 2
$$
Step 2: Perform the multiplications:
$$
2 \times 7 = 14
$$
$$
5 \times 2 = 10
$$
Substitute these results back:
$$
14 + 18 + 10
$$
Step 3: Add the values together:
$$
14 + 18 = 32
$$
Then add $10$:
$$
32 + 10 = 42
$$
Thus, the final result of the expression is $42$.

Key Concept: Grouping of terms doesn’t affect sum, Associative property

c) Assertion is true, but Reason is false.
[Solution Description] According to the associative property of addition, the way in which numbers are grouped does not affect their sum. In this case:
1. Calculate $((-5) + 12) + (-7)$:
$(-5) + 12 = 7$
Then,
$7 + (-7) = 0$
2. Calculate $(-5) + (12 + (-7))$:
$12 + (-7) = 5$
Then,
$(-5) + 5 = 0$
As shown, both groupings result in a sum of zero, confirming the assertion that grouping does not affect the sum. The reason given is incorrect because changing the order in addition does not affect the sum due to the commutative property.
Thus, the correct option is: Assertion is true, but Reason is false.

Key Concept: Expression as Sum, Associative Property

a) -1
[Solution Description]
Let’s apply the associative property for evaluating the expression by changing the grouping of terms.
1. First, add the negative numbers:
$(-6) + (-10) = -16$
2. Then, add this result to $15$:
$-16 + 15 = -1$
Therefore, the final result of the expression is $-1$.

Key Concept: Associative Property

c) $3 + (4 + 5)$
[Solution Description] According to the associative property, altering the grouping of the terms in a sum does not change the result. The expression $3 + (4 + 5)$ can be rearranged as $(3 + 4) + 5$, and both will provide the same sum. Calculating both gives:
$3 + (4 + 5) = 3 + 9 = 12$
$(3 + 4) + 5 = 7 + 5 = 12$
Thus, the sum remains 12 for both groupings.

Key Concept: Grouping of terms doesn’t affect sum

c) $5 + 8 + (-10)$
[Solution Description]
Use the associative property to check equivalency.
Original evaluation:
$8 + (-10) + 5$
Start by computing $8 + (-10)$:
$8 – 10 = -2$
Then add:
$-2 + 5 = 3$
Verifying options in alignment with the associative property:
Option c):
$5 + 8 + (-10)$
Commence with $5 + 8$:
$5 + 8 = 13$
Proceed by adding $-10$:
$13 – 10 = 3$
Resultant sum aligns with prior calculations.

Key Concept: Swapping and Grouping Terms

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion discusses the effect of changing the order in addition expressions like $3 + 5 + 7$. According to the commutative property, which is mentioned in the Reason, swapping terms in addition doesn’t affect the sum. Thus, the Reason correctly explains why the Assertion is true. Therefore, both statements are true, and Reason is the correct explanation of Assertion.

Key Concept: Understanding expressions as the sum of their terms

c) 19
[Solution Description]
To solve this expression using the property that order and grouping do not affect the sum, we can rearrange the terms for easier computation.
Start with the expression:
$8 + 15 – 7 + 3$
We can group $8$ and $3$, and $15$ and $-7$:
$(8 + 3) + (15 – 7)$
Calculate each group:
$8 + 3 = 11$
$15 – 7 = 8$
Add the results of these groups:
$11 + 8 = 19$
Thus, the value of the expression is $19$.

Key Concept: Using associative property of addition

a) $(9 + 4) + 2 = 15$
[Solution Description] According to the associative property of addition, how terms are grouped does not affect their sum. Let’s evaluate:
1. Original expression:
$9 + 4 + 2 = 15$
2. Grouping the first two terms gives:
$(9 + 4) + 2 = 13 + 2 = 15$
The result remains unchanged regardless of grouping due to the associative property.

Key Concept: Commutative Property of Addition

c) 50
[Solution Description]
The commutative property states that numbers can be added in any order without changing the sum. We’ll rearrange $25 + 9 + 16$ to make calculation easier.
Begin with:
$25 + 9 + 16$
Rearrange the terms to facilitate addition:
$25 + 16 + 9$
Compute the first two terms:
$25 + 16 = 41$
Add the last term:
$41 + 9 = 50$
Hence, the simplified result is $50$.

Key Concept: Commutative and Associative Properties, Application to Real-Life Scenarios

a) 25
[Solution Description]
First, we’ll remove the brackets in the expression $15 + (25 – 10) – 5$:
$15 + 25 – 10 – 5$
Using the associative property, let’s group the addition terms:
$(15 + 25) + (-10 – 5)$
Calculate each grouped term:
$15 + 25 = 40$
and
$-10 – 5 = -15$
Adding these results:
$40 + (-15) = 25$
Thus, the final result is $25$.

Key Concept: Swapping and Grouping Terms: Application of Associative Property

c) 10
[Solution Description]
The associative property allows us to change the grouping of numbers without changing their sum. In this case, we will regroup the terms as follows:
Start with the original expression:
$(12 – 7) + 5$
Using the associative property, regroup the numbers:
$12 + (5 – 7)$
Now, calculate inside the parentheses first:
$5 – 7 = -2$
Then add the remaining number:
$12 + (-2) = 10$
Thus, the evaluated result is 10.

Key Concept: Commutative Property, Token Model of Integers

c) 0
[Solution Description]
Rearranging the expression $(-12) + 8 + (-3) + 7$ to make computation easier involves grouping positive and negative numbers together:
$[8 + 7] + [(-12) + (-3)]$
Calculate each set of terms:
$8 + 7 = 15$
and
$-12 + (-3) = -15$
Now add them together:
$15 + (-15) = 0$
Therefore, the sum is $0$.

Key Concept: Swapping and Grouping Terms, Removing Brackets in Expressions

c) 50
[Solution Description]
To simplify the expression $50 – (30 – 20) + 10$, we first need to remove the brackets by changing the signs accordingly:
$50 – 30 + 20 + 10$
Now, we can group the terms using the associative property:
$(50 + 20) + (-30 + 10)$
Calculating each part separately, we have:
$50 + 20 = 70$
and
$-30 + 10 = -20$
Adding these results together gives:
$70 – 20 = 50$
So, the simplified value is $50$.

Key Concept: Associative Property

a) $(x + y) + z = x + (y + z)$
[Solution Description] The associative property of addition states that when three or more numbers are added, the grouping of the numbers does not change the sum. This can be seen in $(x + y) + z = x + (y + z)$.

Key Concept: Swapping and Grouping Terms: Reasoning using Token Model

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
We will examine both the assertion and reason separately to determine their validity and relationship.
Assertion: When adding integers, changing the order of terms does not affect the sum. According to the commutative property of addition, in arithmetic, changing the order of operands does not change the result. Therefore, the assertion is true.
Reason: The Token Model shows that each integer can be represented by tokens, and swapping tokens does not change the total value. In the Token Model used for integers, each positive number is represented by a certain number of positive tokens, and negative numbers are represented by negative tokens. Swapping these tokens or changing their order does not affect the overall count of tokens, thus not altering the sum. Therefore, the reason is also true.
Since both the assertion and reason are true, and the reason provides an explanation for why changing the order of terms does not affect the sum when using the Token Model, option a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion is correct.

Key Concept: Token Model with Multiple Operations

a) 8 – 7 + 5 – 3
[Solution Description]
To transform this expression, resolve the inner operations first:
Inside the bracket:
$$7 – 5 + 3 = 2 + 3 = 5$$
Substitute back into the expression:
$$8 – 5 = 3$$
Apply the token model to swap signs when removing brackets:
$8 – (7 – 5 + 3) = 8 – 7 + 5 – 3$
Simplify:
$$8 – 7 + 5 – 3 = 1 + 5 – 3 = 6 – 3 = 3$$
Therefore, the transformed expression is $8 – 7 + 5 – 3$.

Key Concept: Commutative and Associative Properties of Addition

c) $((-12 – 5) + 15) + 8$
[Solution Description]
We begin with the expression $(-12 + 8) + (15 – 5)$. First, convert the subtraction to addition by taking the inverse:
$(-12 + 8) + (15 + (-5))$
Now apply the commutative property by swapping $8$ and $-5$:
$(-12 – 5) + (15 + 8)$
Then, using the associative property, group the new pairs:
$((-12 – 5) + 15) + 8$
Evaluating gives:
$(-17 + 15) + 8 = -2 + 8 = 6$
Thus, the correct rearrangement is $((-12 – 5) + 15) + 8$.

Key Concept: Swapping and Grouping, Commutative Property

b) You can swap the order since it doesn’t affect the result.
[Solution Description]
The commutative property states that the order of operations can be swapped without changing the outcome. For writing a letter and reading a book, the result is the same regardless of which task is done first. Therefore, this scenario demonstrates the commutative property as swapping these tasks will not change the end result.

Key Concept: Understanding Swapping Order

c) Either way works
[Solution Description] The order in which Maria puts on her jacket and hat does not matter. She will look the same whether she wears the jacket first or the hat first.

Key Concept: Swapping and Grouping in Everyday Life

b) Preheating should come first to ensure the oven is hot when the batter is ready.
[Solution Description] For baking, it is essential to preheat the oven first so that when the batter is ready, the oven is already at the right temperature for baking. Mixing the batter first would mean waiting for the oven to reach the required temperature, potentially affecting the texture or rise of the cake.

Key Concept: Swapping and Grouping in Everyday Life

b) No, because both actions are unrelated and do not depend on one another.
[Solution Description] The order does not matter in this scenario because brushing teeth and putting on a backpack are independent activities that can be done without affecting each other. John’s readiness for school will remain the same regardless of which action he does first.

Key Concept: Creating Expressions, Multi-Step Calculation

c) 38
[Solution Description]
The problem requires us to calculate the total number of students based on grouping and additional helpers.
First, calculate the total number of students in the groups: $5$ groups $\times 7$ students per group = $35$ students.
Next, add the extra $3$ students who are helping: Total = $35 + 3 = 38$.
Thus, the complete expression representing this scenario is $5 \times 7 + 3$.
Identifying the terms: $5 \times 7$ and $3$ are the two terms of the expression.
Hence, the answer is 38.

Key Concept: Expression with Division and Addition

a) $\frac{60}{12} + 2$
[Solution Description]
To determine the number of new packets from the remaining candies, we divide the total candies by the number of candies per packet: $\frac{60}{12} = 5$. She already has 2 packets, so the expression for the total packets is $5 + 2 = 7$. Therefore, the correct expression is $\frac{60}{12} + 2$.
The terms in the expression are $\frac{60}{12}$ and $2$.

Key Concept: Using Expressions to Calculate Costs

a) $5 \times 30 + 20$
[Solution Description]
The calculation for the cost of the pasta is done by multiplying the number of plates by the cost per plate: $5 \times 30 = 150$. Including the service charge leads to $150 + 20 = 170$. Thus, the expression representing their total bill is $5 \times 30 + 20$.
The terms in this expression are $5 \times 30$ and $20$.

Key Concept: Expressions, Terms Identification

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let’s analyze both statements:
In the assertion (A), the expression $6 \times 5 + x$ with $x = 3$ is indeed correctly representing the situation described: “33 students are playing in groups of 5 with a remainder.” The expression $6 \times 5$ gives the number of students that can be fully grouped into groups of 5, which equals 30 students as $6 \times 5 = 30$, leaving 3 out.
On examining Reason (R), it states that $6 \times 5 + 3$ indicates 6 groups of 5 plus another group of 3. This explanation aligns with the stated situation because after forming full groups of 5 via $6 \times 5$, there remain 3 students who are not part of any complete group, represented by “+3”.
Therefore, while both Assertion and Reason are true, the Reason directly explains why the Assertion is structured as it is; hence, Reason is the correct explanation of Assertion.