Variables and algebraic symbols: 48 Practice MCQs with Answers and Explanation

Q1. A school club charges a fixed registration fee of $150$ rupees plus $40$ rupees for each workshop attended. Which expression represents the total cost for a student attending $w$ workshops, and what does $40w$ represent?

A.

150w+40

; the registration cost

B.

150+40w

; the workshop cost ✅

C.

190w

; the total fixed cost

D.

150+40

; the number of workshops

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The fixed registration fee is paid once, so it is represented by $150$ . Each workshop adds $40$ rupees, so $w$ workshops cost $40w$ . Therefore, $150+40w$ correctly models the situation.

Q2. A student writes $3x+5=20$ to represent a number increased by $5$ , then multiplied by $3$ , but the intended situation is that the number is first increased by $5$ and the result is multiplied by $3$ . Which equation correctly represents the situation?

A.

3x+5=20

B.

3(x+5)=20

✅

C.

x+15=20

D.

3(x-5)=20

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Parentheses are essential because the entire quantity $x+5$ must be multiplied by $3$ . The expression $3x+5$ multiplies only $x$ , which changes the order of operations and models a different situation.

Q3. A delivery service charges $80$ rupees for the first kilometer and $25$ rupees for every additional kilometer. If $d$ is the total number of kilometers traveled, which expression correctly models the cost for $d\ge1$ ?

A.

80+25d

B.

80d+25

C.

80+25(d-1)

✅

D.

105d

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The first kilometer is already included in the $80$ -rupee charge. Only the remaining $d-1$ kilometers receive the $25$ -rupee charge, giving $80+25(d-1)$ . This avoids charging the first kilometer twice.

Q4. A graph shows a straight line passing through $(0,50)$ and $(4,130)$ , where $x$ represents hours and $y$ represents total savings. Which algebraic expression best represents the relationship?

A.

y=20x+50

✅

B.

y=50x+20

C.

y=80x+50

D.

y=20x+130

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The initial value is $50$ , so the constant term is $50$ . The rate is $(130-50)/(4-0)=20$ , meaning savings increase by $20$ per hour. Thus, $y=20x+50$ .

Q5. A student claims that $4(a+3)$ and $4a+3$ are equivalent because the $4$ is next to both expressions. Which statement best evaluates the student's reasoning?

A.The student is correct because multiplication affects only the first term

B.The student is incorrect because

4(a+3)=4a+12

✅

C.The student is correct because

4(a+3)=4a+3

D.The student is incorrect because

4(a+3)=a+12

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The factor $4$ multiplies every term inside the parentheses. Therefore, $4(a+3)=4a+12$ . The student's mistake is failing to distribute the multiplier to both $a$ and $3$ .

Q6. Two students model the number of pencils in $n$ identical boxes. Student A writes $6n$ , while Student B writes $n+6$ . Each box contains $6$ pencils. Which conclusion is most accurate?

A.Student A is correct because

6

is multiplied by the number of boxes ✅

B.Student B is correct because

6

pencils are added once

C.Both are always equivalent

D.Neither expression uses a variable correctly

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: If each of $n$ boxes contains $6$ pencils, multiplication is required: $6\times n=6n$ . The expression $n+6$ would describe adding six to the number of boxes, which is not the intended situation.

Q7. A rectangular garden has length $x+4$ meters and width $x$ meters. A student says its perimeter is $2x+4$ . Which expression gives the correct perimeter and best explains the error?

A.

2x+4

, because only the different dimensions are added

B.

x^2+4x

, because perimeter requires multiplication

C.

4x+8

, because

2[(x+4)+x]=4x+8

✅

D.

2x+8

, because only the constant should be doubled

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Perimeter is twice the sum of the length and width: $2[(x+4)+x]$ . Simplifying gives $4x+8$ . The student's $2x+4$ fails to double both dimensions and therefore underestimates the perimeter.

Q8. A school charges a fixed registration fee of $500$ rupees plus $120$ rupees for each workshop attended. If $w$ represents the number of workshops, which expression correctly models the total cost, and why?

A.

500w+120

B.

120w+500

✅

C.

620w

D.

500+120

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The workshop charge changes with the number of workshops, so $120$ is multiplied by the variable $w$ . The registration fee remains unchanged, making $500$ the constant. Therefore, $120w+500$ correctly models the situation.

Q9. A student claims that in $7x+14$ , both $7$ and $14$ are variables because they are numbers appearing in an algebraic expression. Which response best evaluates the student's reasoning?

A.Both numbers are variables because they affect the expression.

B.

7

is the variable and

14

is the constant.

C.

x

is the variable, while

7

and

14

are constants. ✅

D.Only

14

is a constant because it is the larger number.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The symbol $x$ can represent changing values, so it is the variable. The numbers $7$ and $14$ have fixed values in the expression, making them constants. Their numerical size does not determine whether they are variables.

Q10. A taxi company models a fare using $F=80+25d$ , where $d$ is the distance traveled in kilometers. If the distance changes from $4$ km to $9$ km, which statement correctly identifies what changes and calculates the new fare?

A.Only

80

changes, and the fare becomes

305

.

B.Only

25

changes, and the fare becomes

225

.

C.

d

changes, while

80

and

25

remain fixed; the new fare is

305

. ✅

D.All three quantities change because the fare changes.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The variable $d$ represents distance and changes from $4$ to $9$ . The values $80$ and $25$ are constants in this model. Substituting $d=9$ gives $F=80+25(9)=305$ , so option C is correct.

Q11. A learner evaluates $3n+40$ for $n=6$ and writes $3(6)+40=58$ . Another learner says the answer should be $3+6+40=49$ because every symbol and number must be added. Which analysis is correct?

A.The first learner is correct because multiplication must be performed before addition. ✅

B.The second learner is correct because variables always represent addition.

C.Both learners are correct because

3n

can mean either

3+n

or

3\times n

.

D.Neither learner is correct because

40

should be treated as a variable.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The notation $3n$ means $3\times n$ , not $3+n$ . Substituting $n=6$ gives $3\times6+40=18+40=58$ . The second learner has confused multiplication with addition, a common algebraic misconception.

Q12. A graph shows the points $(0,50)$ , $(1,70)$ , $(2,90)$ , and $(3,110)$ on a straight line. If $x$ represents the number of items and $y$ represents total cost, which interpretation is most reasonable?

A.

50

is the variable cost per item and

20

is the constant.

B.

20

is the variable cost per item and

50

is the constant starting cost. ✅

C.Both

20

and

50

are variables because the graph rises.

D.

x

is constant while

y

is the variable.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The graph begins at $y=50$ when $x=0$ , showing a fixed starting amount of $50$ . Each increase of one item raises the cost by $20$ , so $20$ represents the changing rate while $50$ remains constant.

Q13. Two expressions are proposed for the cost of renting bicycles: Model A is $300+75h$ , while Model B is $375h$ , where $h$ is the number of hours. For a one-hour rental both give the same cost. Which reasoning best explains why Model A is more informative for multiple hours?

A.Model B is always better because it contains only multiplication.

B.Model A separates a fixed cost from a cost that depends on hours, while Model B incorrectly makes the entire cost proportional to

h

. ✅

C.Both models are equivalent for every value of

h

.

D.Model A has no variable because

300

is present.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Although both models produce $375$ when $h=1$ , they behave differently for other values. Model A includes a fixed $300$ -rupee charge plus $75$ per hour. Model B incorrectly assumes the entire $375$ changes with every hour.

Q14. A number $k$ is selected so that the expression $5k+12$ has the same value as $47$ . A student argues that $12$ must also change because otherwise the expression cannot reach $47$ . Which conclusion correctly challenges this argument?

A.The constant

12

stays fixed; solving

5k+12=47

gives

k=7

. ✅

B.The constant

12

changes to

7

, while

k

remains fixed.

C.Both

5

and

12

must change to make the expression equal

47

.

D.The expression cannot equal

47

because constants never participate in calculations.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: A constant can participate in an operation without changing its own value. Subtracting $12$ from both sides gives $5k=35$ , and dividing by $5$ gives $k=7$ . Thus only the variable $k$ needs a suitable value.

Q15. A student sees $18-3\times4$ and argues that the answer is $60$ because subtraction should be performed before multiplication. Which reasoning correctly evaluates the expression?

A.Subtract first:

18-3=15

, then multiply to get

60

B.Multiply first:

3\times4=12

, then subtract from

18

to get

6

✅

C.Add the numbers first, then multiply to get

84

D.Divide

18

by

3

, then subtract

4

to get

2

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The multiplication operation is evaluated before subtraction when no grouping symbols change the order. Therefore, $3\times4=12$ , and then $18-12=6$ . The student's error comes from reversing the usual operation priority.

Q16. A shop charges a fixed delivery fee of $120$ rupees plus $35$ rupees for each item. If a customer orders $6$ items, which expression correctly models the total cost?

A.

120+35+6

B.

120\times35+6

C.

120+35\times6

✅

D.

120-35\times6

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The fixed fee is $120$ , while the item cost is $35$ multiplied by $6$ , giving $35\times6=210$ . Adding the fixed fee gives $120+210=330$ rupees, so $120+35\times6$ is the correct model.

Q17. A student simplifies $24\div6+2$ as $24\div8=3$ by combining the $6$ and $2$ first. What is the best analysis of the student's mistake?

A.Division should be replaced by subtraction

B.Addition must be performed after division here, so the correct value is

4+2=6

✅

C.The expression must always equal

24

D.The student should multiply

6

and

2

first, giving

24\div12=2

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The student incorrectly grouped $6+2$ even though no grouping symbols require that step. Division is performed before addition, so $24\div6=4$ , followed by $4+2=6$ . This error is caused by inventing an unsupported grouping.

Q18. A graph shows four points on a number line representing $x-3$ , $x+3$ , $3-x$ , and $x\times3$ . For $x=2$ , which expression corresponds to the point located at $-1$ ?

A.

x+3

B.

x\times3

C.

3-x

D.

x-3

✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Substituting $x=2$ gives $x-3=2-3=-1$ . The other expressions give $5$ , $6$ , and $1$ , respectively. Interpreting the position on the number line therefore identifies $x-3$ as the matching expression.

Q19. A rectangle has a length represented by $x+4$ and a width represented by $3$ . A student writes its perimeter as $x+4+3$ . Which expression correctly represents the perimeter, and why?

A.

x+7

, because only the distinct sides are counted

B.

2(x+4+3)

, because both pairs of equal sides are included ✅

C.

3(x+4)

, because multiplication always represents perimeter

D.

2(x+4)-3

, because the width is subtracted

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A rectangle has two lengths and two widths. Therefore, the perimeter is $2(x+4)+2(3)$ , which can be written as $2(x+4+3)$ . The student's expression counts only one length and one width, giving half the perimeter.

Q20. A number machine follows the rule: start with a number, multiply it by $4$ , subtract $6$ , then divide the result by $2$ . Which single expression represents the machine's output for an input $x$ , and what output does it produce when $x=5$ ?

A.

4x-\frac{6}{2}=17

B.

4(x-6)\div2=-2

C.

(4x-6)\div2=7

✅

D.

4(x-6\div2)=8

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Following the operations in the stated sequence gives $4x$ , then $4x-6$ , and finally $(4x-6)\div2$ . For $x=5$ , this becomes $(20-6)\div2=14\div2=7$ , so option C correctly models both the process and result.

Q21. A student writes $3x+5=20$ and says the symbol $=$ means “calculate the expression on the left.” Which interpretation is mathematically more accurate?

A.It means the left side must always be simplified first.

B.It states that the expressions on both sides have the same value. ✅

C.It means the number on the right is always the answer.

D.It means the left expression is greater than or equal to the right.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The equality symbol states that two expressions have the same value. It does not simply mean calculate, and neither side is automatically an answer. Both sides represent equivalent quantities when the statement is true.

Q22. A teacher models a balance scale with $2x+4$ units on the left and $14$ units on the right. Which equation correctly represents the balance condition?

A.

2x=14+4

B.

2x-4=14

C.

2x+4=14

✅

D.

14+4=2x

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A balanced scale means both quantities have equal values. Therefore, the complete left quantity $2x+4$ must equal the complete right quantity $14$ . The equality symbol connects these two equivalent quantities.

Q23. A shop charges a fixed fee of $50$ rupees plus $20$ rupees for each notebook. Another shop charges $30$ rupees per notebook. For how many notebooks will the total costs be equal?

A.

x=3

B.

x=5

✅

C.

x=10

D.

x=15

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The two pricing models can be represented by $50+20x=30x$ . Subtracting $20x$ gives $50=10x$ , so $x=5$ . At five notebooks, both models produce the same total cost.

Q24. A student solves $4x+8=20$ by writing $4x=20+8$ , then $4x=28$ . What is the main error in the reasoning?

A.The student should divide by

8

before subtracting.

B.The

8

should be subtracted from both sides, not added. ✅

C.The equality symbol cannot be used with variables.

D.The student should multiply both sides by

8

.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: To preserve equality, the same operation must be applied appropriately to both sides. Since $8$ is added to $4x$ , subtracting $8$ from both sides gives $4x=12$ . Adding $8$ changes the balance.

Q25. A graph shows two lines representing $y=2x+6$ and $y=4x$ . At their intersection, the two expressions have equal values. What does the intersection represent?

A.A point where both expressions have the same output value. ✅

B.A point where both lines have zero slope.

C.A point where the first line is always greater.

D.A point where

x

and

y

must both equal zero.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: At an intersection, both equations produce the same $y$ -value for the same $x$ -value. Thus the equality represented there identifies a common solution to the two expressions rather than merely indicating that either expression equals zero.

Q26. Two students compare the statements $7+5=12$ and $7+5=13$ . Student A says both are equations because they contain $=$ . Student B says only one is a true equality. Who is correct, and why?

A.Student A, because every statement containing

=

is true.

B.Student B, because an equality statement must have equal values on both sides. ✅

C.Both are correct because equations never need to be true.

D.Neither is correct because

=

is only used with variables.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Student B is correct. The first statement has equal values, since $7+5=12$ , while the second does not because $7+5$ equals $12$ , not $13$ . The symbol asserts equality, so the assertion must be true.

Q27. Find the value of $x$ that makes $2(x+3)=x+11$ true. Then explain why the equality symbol is essential in this situation.

A.

x=3

, because both sides become

12

.

B.

x=5

, because both sides become

16

. ✅

C.

x=7

, because both sides become

20

.

D.

x=9

, because both sides become

24

.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Expanding gives $2x+6=x+11$ . Subtracting $x$ from both sides and then $6$ gives $x=5$ . Substitution confirms both sides equal $16$ , showing why equality identifies the required balance between expressions.

Q28. A temperature-control system keeps a room comfortable when the temperature $T$ is at least $18^\circ C$ but below $24^\circ C$ . Which inequality represents the acceptable range?

A.

18<T<24

B.

18\leq T<24

✅

C.

18<T\leq24

D.

18\leq T\leq24

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The phrase 'at least $18^\circ C$ ' includes $18$ , so the first inequality must use $\leq$ . The phrase 'below $24^\circ C$ ' excludes $24$ , requiring $<$ . Therefore, $18\leq T<24$ correctly models the situation.

Q29. A student claims that $x>5$ means $x$ can equal $5$ because numbers greater than $5$ begin at $5$ . What is the precise flaw in this reasoning?

A.The symbol

>

includes the boundary value

B.The symbol

>

excludes the boundary value ✅

C.The symbol

>

means less than or equal to

D.The symbol

>

represents equality only

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The student's mistake is treating the boundary as part of the solution. The symbol $>$ means strictly greater than, so $5$ itself is excluded. Values such as $5.1$ , $6$ , and $10$ satisfy $x>5$ , but $5$ does not.

Q30. A delivery company charges a fee when a package weighs more than $10$ kg and accepts packages weighing up to $25$ kg. Which condition describes packages that are accepted and charged this way?

A.

10\leq w\leq25

B.

10<w\leq25

✅

C.

10<w<25

D.

10\leq w<25

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The package must weigh more than $10$ kg, so $10$ is excluded and the symbol is $>$ . It can weigh up to $25$ kg, meaning $25$ is included with $\leq$ . Thus $10<w\leq25$ captures both conditions.

Q31. On a number line, an open circle is placed at $3$ and the shaded ray extends to the left. Which inequality does the graph represent, and why?

A.

x\geq3

, because the ray points left

B.

x>3

, because the circle is open

C.

x<3

, because the circle is open and shading is left ✅

D.

x\leq3

, because the ray points left

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: An open circle means the boundary value $3$ is not included. Shading to the left represents numbers smaller than $3$ . Combining these two observations gives $x<3$ , not $x\leq3$ , because $3$ itself is excluded.

Q32. A school allows students to enter a competition if their score $s$ is greater than or equal to $70$ . One student says $s>70$ is equivalent because both conditions describe high scores. Which score proves the student's claim is incorrect?

A.69

B.70 ✅

C.71

D.85

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The condition $s\geq70$ includes $70$ , while $s>70$ excludes it. Therefore, a score of exactly $70$ satisfies the school's requirement but would not satisfy the student's proposed inequality. This boundary-value difference makes the two statements unequal.

Q33. A machine operates safely when pressure $p$ is no greater than $90$ units and greater than $40$ units. An operator writes $40\leq p\leq90$ . Which part of the model is incorrect?

A.The lower boundary should be excluded ✅

B.The upper boundary should be excluded

C.Both boundaries should be excluded

D.No part is incorrect

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: 'Greater than $40$ ' means $40$ is not allowed, so the lower condition must be $p>40$ . 'No greater than $90$ ' means $90$ is allowed, giving $p\leq90$ . The correct model is $40<p\leq90$ , so the operator incorrectly included $40$ .

Q34. Two students compare the statements $a\leq7$ and $b<7$ . They argue that the first always describes more possible values because it contains an equality symbol. Which conclusion is mathematically correct?

A.The first always has fewer solutions

B.The second always has more solutions

C.They differ only at the boundary value

7

✅

D.They represent exactly the same set of values

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Both conditions include every number below $7$ , but they treat $7$ differently. The statement $a\leq7$ includes $7$ , whereas $b<7$ excludes it. Thus, the difference is specifically the boundary value, not the entire range of solutions.

Q35. A student evaluates $18-[6-(2+5)]$ as $18-6-2+5=15$ . Which statement best identifies the error and gives the correct value?

A.The student ignored the parentheses; the value is 13.

B.The student changed the subtraction inside the brackets; the value is 25.

C.The student removed grouping symbols without preserving their effect; the value is 19. ✅

D.The student should evaluate the innermost grouping first; the value is 25.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The innermost grouping is $2+5=7$ . Then $6-7=-1$ , so the entire expression becomes $18-(-1)=19$ . Therefore the student's expansion is incorrect, and the correct value is 19, making option C correct rather than D.

Q36. A rectangular garden has $x$ meters of length and $x+4$ meters of width. A path of $2$ meters is added to both dimensions. Which expression correctly represents the new area?

A.

x(x+4)+2

B.

(x+2)(x+6)

✅

C.

2x+6

D.

x(x+4+2)

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The original dimensions are $x$ and $x+4$ . Adding $2$ meters to each dimension changes them to $x+2$ and $x+6$ . Multiplying these new dimensions gives $(x+2)(x+6)$ , so B correctly models the situation.

Q37. Consider the expression $24\div[3(2+2)]$ . Two students disagree. Student A gets $2$ , while Student B gets $16$ . Which evaluation is correct, and why?

A.Student A, because division is performed before multiplication inside the brackets.

B.Student B, because

3(2+2)=12

, so

24\div12=2

. ✅

C.Neither; the correct value is

8

because multiplication must always come before division.

D.Neither; the correct value is

6

because the brackets should be removed first.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: First evaluate the parentheses: $2+2=4$ . Then the bracket contains $3(4)=12$ . Finally, $24\div12=2$ . Student B correctly preserves the grouping structure and evaluates the expression from the inside outward.

Q38. A number-line model shows all values satisfying $x$ greater than $3$ and less than or equal to $8$ . Which expression correctly represents the interval shown?

A.

3\le x<8

B.

3<x\le8

✅

C.

3<x<8

D.

3\le x\le8

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Because $3$ is excluded, the inequality at $3$ must be strict: $3<x$ . Because $8$ is included, the inequality at $8$ must use $\le$ . Thus the matching expression is $3<x\le8$ , making B correct.

Q39. A teacher writes $5-\{2-[3-(x+1)]\}$ . A student claims that replacing every grouping symbol with parentheses gives a different expression. Which response is most accurate?

A.The student is correct because braces always change numerical values.

B.The student is correct because brackets must always be evaluated before parentheses.

C.The student is incorrect because grouping symbols can organize nested operations without changing the intended arithmetic. ✅

D.The student is incorrect only when

x

is positive.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Parentheses, brackets, and braces can all serve as grouping symbols. Their different shapes mainly help readers distinguish nested levels. Replacing the shapes consistently with parentheses preserves the intended structure and therefore does not automatically change the expression's value.

Q40. A learner simplifies $7-\{3-[2-(x-4)]\}$ to $7-3-2-x-4$ . Which change would best correct the reasoning?

A.Distribute the negative signs through each grouping level carefully before combining terms. ✅

B.Remove all grouping symbols first and then combine like terms.

C.Change every subtraction sign to addition because nested groups reverse operations.

D.Evaluate

x-4

as

x+4

before removing the parentheses.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The learner loses track of sign changes when crossing nested grouping levels. First simplify the innermost group $2-(x-4)$ , then account for the negative sign before the bracket, and finally the negative sign before the braces. Careful distribution preserves each sign correctly.

Q41. For the expression $12-[4-\{2+(x-3)\}]$ , a student says that changing the braces to parentheses makes the expression invalid because braces have a special arithmetic operation. Which conclusion is strongest?

A.The student is correct because braces always mean multiplication.

B.The student is correct because only parentheses can contain variables.

C.The student is incorrect because braces, brackets, and parentheses can all indicate grouping; their shape mainly clarifies nesting. ✅

D.The student is incorrect only when the expression contains subtraction.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Grouping symbols do not represent separate arithmetic operations merely because their shapes differ. Parentheses, brackets, and braces can all indicate that a set of terms should be treated together. Different shapes are especially useful when several levels of grouping are nested.

Q42. A school club charges a fixed registration fee of $120$ rupees plus $35$ rupees for each meeting attended. Which equation represents the total cost $C$ for $m$ meetings, and what is the cost for $4$ meetings?

A.

C=120+35m

, so

C=260

✅

B.

C=155m

, so

C=620

C.

C=35+120m

, so

C=515

D.

C=120m+35

, so

C=515

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The fixed fee is paid once, so it is the constant $120$ , while $35$ is multiplied by the number of meetings. Substituting $m=4$ gives $120+35(4)=260$ . The distractors reverse the roles of the fixed and variable quantities.

Q43. Two students simplify $3(x+4)-2x$ . Student A obtains $x+12$ , while Student B obtains $x+4$ . Which evaluation best explains the disagreement?

A.Student A is correct because

3

must be multiplied only by

4

B.Student B is correct because

3x-2x=x

and the

12

cancels ✅

C.Both are correct because equivalent expressions can have different constant terms

D.Neither is correct because the simplified expression should contain

2x

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Distributing $3$ across the parentheses gives $3x+12$ . Combining like terms then gives $3x-2x+12=x+12$ . Therefore Student A is correct. Student B correctly combines the variable terms but incorrectly drops the constant $12$ .

Q44. A taxi company models a fare with $F=80+25k$ , where $k$ is the number of kilometers traveled. A passenger has $230$ rupees. What is the greatest whole number of kilometers the passenger can travel without exceeding the budget?

A.5 km

B.6 km ✅

C.7 km

D.8 km

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Set the fare equal to the available budget: $80+25k\le230$ . Subtracting $80$ gives $25k\le150$ , so $k\le6$ . Thus $6$ kilometers is the greatest whole-number distance. Choosing $7$ would produce a fare of $255$ rupees.

Q45. A student solves $4x+7=23$ by subtracting $7$ and writing $4x=23$ , then concludes $x=23/4$ . What is the student's key error?

A.The student should divide by

7

first

B.The student forgot to subtract

7

from

23

✅

C.The student should multiply

4

by

7

before solving

D.The student should subtract

4

from both sides

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The constant $7$ must be removed from both sides, giving $4x=16$ , not $4x=23$ . Then dividing by $4$ gives $x=4$ . The error is failing to apply the subtraction to the numerical side of the equation.

Q46. A graph of $y=2x+1$ and a horizontal line $y=9$ intersect at one point. Without calculating every value, which ordered pair must represent their intersection?

A.

(3,9)

B.

(4,9)

✅

C.

(5,9)

D.

(9,4)

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: At the intersection, both expressions have the same $y$ -value. Setting $2x+1=9$ gives $2x=8$ , so $x=4$ . Therefore the intersection is $(4,9)$ . The distractors reflect common confusion between the slope, intercept, and coordinate order.

Q47. A rectangular garden has length $x+5$ meters and width $x$ meters. Its perimeter is $34$ meters. Which equation correctly models the situation and gives the garden's dimensions?

A.

x+(x+5)=34

, giving

x=14.5

B.

2x+5=34

, giving

x=14.5

C.

2[x+(x+5)]=34

, giving

x=6

✅

D.

x(x+5)=34

, giving

x=4

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Perimeter requires adding all four sides, so $2[x+(x+5)]=34$ . Simplifying gives $4x+10=34$ , hence $4x=24$ and $x=6$ . The dimensions are $6$ m by $11$ m. The other choices confuse perimeter with a partial sum or area.

Q48. A number $x$ satisfies $3(x-2)+4=2x+9$ . A student claims there is no solution because the coefficients of $x$ on the two sides are different. Which conclusion is correct?

A.There is no solution because

3\ne2

B.There is exactly one solution,

x=11

✅

C.There is exactly one solution,

x=7

D.Every real number is a solution

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Expanding gives $3x-6+4=2x+9$ , so $3x-2=2x+9$ . Subtracting $2x$ gives $x-2=9$ , hence $x=11$ . The student's reasoning is incorrect because different coefficients do not prevent an equation from having a unique solution.

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