"Mathematics — Algebra" — Practice

Question 1

Simplify: 7a − 4b − 3a + 9b.
(a) 4a + 5b  (b) 4a − 5b  (c) 10a + 5b  (d) 4a + 13b

Accepted Answer

Group like terms: (7a−3a)+(−4b+9b) = 4a+5b.

Answer: (a) 4a+5b. Trap (b) 4a−5b comes from a sign error combining the b-terms, treating −4b+9b as −5b instead of the correct +5b. Trap (c) 10a+5b comes from a sign error combining the a-terms, wrongly adding 7a+3a=10a instead of correctly subtracting (7a−3a=4a). Trap (d) 4a+13b comes from combining −4b and +9b as if both were positive, adding their magnitudes (4+9=13) instead of correctly finding the signed sum.

Question 2

Multiply: 4x²y × (−3xy²).
(a) −12x³y³  (b) 12x³y³  (c) −12x²y²  (d) x³y³

Accepted Answer

Multiply coefficients: 4×(−3)=−12. Add exponents of matching variables: x^(2+1)=x³, y^(1+2)=y³. Result: −12x³y³.

Answer: (a) −12x³y³. Trap (b) 12x³y³ comes from a sign error, forgetting that a positive coefficient times a negative coefficient gives a negative result. Trap (c) −12x²y² comes from multiplying the exponents instead of adding them (2×1=2, 1×2=2), rather than correctly summing them. Trap (d) x³y³ comes from adding the coefficients instead of multiplying them (4+(−3)=1), effectively dropping the coefficient's correct magnitude and sign.

Question 3

Using a standard algebraic identity, find the value of 102² without direct long multiplication.
(a) 10404  (b) 10004  (c) 10204  (d) 9604

Accepted Answer

102=100+2. Using (a+b)²=a²+2ab+b² with a=100, b=2: 100²+2(100)(2)+2² = 10000+400+4 = 10404.

Answer: (a) 10404. Trap (b) 10004 comes from the classic 'freshman's dream' error, dropping the middle term entirely: 100²+2²=10000+4=10004. Trap (c) 10204 comes from forgetting the factor of 2 in the middle term, using ab=200 instead of 2ab=400: 10000+200+4=10204. Trap (d) 9604 comes from wrongly applying the (a−b)² identity instead of (a+b)²: 100²−2(100)(2)+2²=10000−400+4=9604 — the correct identity for 98² (=100−2), not 102².

Question 4

The sum of three consecutive integers is 72. Find the largest of the three integers.
(a) 25  (b) 24  (c) 23  (d) 27

Accepted Answer

Let the integers be x, x+1, x+2. Sum: x+(x+1)+(x+2)=3x+3=72 ⇒ 3x=69 ⇒ x=23. The largest integer is x+2=25.

Answer: (a) 25. Trap (b) 24 reports the middle integer (x+1) instead of the largest one actually asked for. Trap (c) 23 reports the smallest integer (x) instead of the largest. Trap (d) 27 comes from a transposition sign error — wrongly computing 3x=72+3=75 instead of 3x=72−3=69, giving x=25 and a (incorrect) largest integer of 27.

Question 5

If a:b = 3:5 and b:c = 4:7, find a:b:c.
(a) 12:20:35  (b) 3:4:7  (c) 12:20:28  (d) 15:20:35

Accepted Answer

The two ratios share the term b, currently written as 5 and 4 respectively. Scale both ratios so b matches at their LCM, 20: multiply a:b=3:5 by 4 to get 12:20; multiply b:c=4:7 by 5 to get 20:35. Combined: a:b:c=12:20:35.

Answer: (a) 12:20:35. Trap (b) 3:4:7 simply writes the two given ratios side by side without scaling b to a common value at all — the values of b (5 and 4) don't even match. Trap (c) 12:20:28 correctly scales a:b to 12:20 but incorrectly scales c using the wrong factor (7×4=28 instead of 7×5=35), applying b's scale factor to the wrong ratio. Trap (d) 15:20:35 correctly scales b:c to 20:35 but incorrectly scales a using the wrong factor (3×5=15 instead of 3×4=12), swapping which scale factor belongs to which original ratio.

Question 6

A car travels at a constant speed. Which of the following correctly describes the relationship between the distance covered (d) and the time taken (t)?
(a) d and t are in direct variation; d/t stays constant  (b) d and t are in inverse variation; d×t stays constant  (c) d and t are unrelated quantities  (d) d and t are in direct variation; d+t stays constant

Accepted Answer

At constant speed, d=speed×t, so d/t=speed, a fixed value — the ratio between corresponding values of d and t stays constant, which is exactly the signature of direct variation.

Answer: (a). Trap (b) misclassifies this as inverse variation, likely by confusing it with the genuinely inverse relationship between speed and time for a FIXED distance — mixing up which pair of quantities is being held fixed. Trap (c) is simply false, ignoring the well-known d=speed×t relationship entirely. Trap (d) correctly identifies direct variation but wrongly names the constant invariant as a sum (d+t) rather than a ratio (d/t), misunderstanding what 'constant ratio' actually means in a direct-variation relationship.

Question 7

A shopkeeper marks up the cost price of an item by 25% and then offers a 20% discount on the marked price. What is the overall result?
(a) No profit, no loss  (b) 5% profit  (c) 5% loss  (d) 45% profit

Accepted Answer

Take cost price (CP) = 100. Marked price after a 25% markup: 100×1.25=125. After a 20% discount on the MARKED price: 125×0.80=100. The selling price equals the original cost price exactly — no net profit or loss.

Answer: (a) No profit, no loss. Trap (b) 5% profit comes from the invalid shortcut of simply subtracting the two percentages (25%−20%=5%), ignoring that each percentage is applied to a different base value (CP for the markup, marked price for the discount). Trap (c) 5% loss makes the same invalid-subtraction error in the opposite direction (20%−25%=−5%). Trap (d) 45% profit comes from simply adding the two percentages (25%+20%=45%) instead of correctly combining them multiplicatively step by step.

Question 8

What is the degree of the algebraic expression 5x³y² + 7x²y⁴ − 3xy?
(a) 6  (b) 5  (c) 7  (d) 2

Accepted Answer

Find the degree of each term separately, then take the highest: 5x³y² has degree 3+2=5; 7x²y⁴ has degree 2+4=6; −3xy has degree 1+1=2. The highest is 6.

Answer: (a) 6. Trap (b) 5 reports only the first term's degree, without checking the remaining terms for a higher one. Trap (c) 7 comes from mixing exponents across different terms — wrongly adding the highest power of x seen anywhere (3, from the first term) to the highest power of y seen anywhere (4, from the second term), rather than computing each term's own degree separately. Trap (d) 2 reports only the last listed term's degree (−3xy), ignoring the other two terms entirely.

Question 9

When presented with the equation 6+9=☐+8, a Class VI student writes 15 in the blank, producing 6+9=15+8. This response most directly indicates that the student is treating the equals sign as:
(a) A symbol for relational equivalence between two sides  (b) An instruction to compute and record a running total, read left to right  (c) A balance that must be preserved on both sides  (d) An operator with the same meaning as a comma

Accepted Answer

The student computed 6+9=15 and simply wrote that result in the blank, without checking whether the resulting statement (6+9=15+8) is actually true — exactly the operational reading of '=' as 'compute the answer,' built from years of a+b=☐ practice.

Answer: (b). Trap (a) describes the CORRECT, target understanding — the relational view — which is the opposite of what this student's error actually demonstrates. Trap (c) also describes the correct balance-based understanding, again the opposite of the operational error shown here. Trap (d) misdescribes the equals sign as a meaningless separator, which doesn't fit the student's behaviour either — they did perform a calculation, just applied it to the wrong side of the equation.

Question 10

Which teaching strategy would most effectively help a Class VII class move from an operational to a relational understanding of the equals sign before formally solving linear equations?
(a) Assign twenty equation-solving problems for independent homework practice.
(b) Use a physical or pictorial balance-scale model, showing both sides of an equation as pans that must stay level, and let students perform matched operations on each side.
(c) Provide a memorised list of transposition rules ('change side, change sign') for students to apply directly.
(d) Skip equations with a variable on both sides until a much later class.

Accepted Answer

The balance-scale model makes the relational meaning of '=' physically concrete — both pans must stay level throughout, giving students a direct, visual reason why performing the same operation on both sides preserves equality, before any symbolic rule is introduced.

Answer: (b). Trap (a) assumes a fluency the class hasn't built yet — drilling equation-solving practice addresses the symptom, not the conceptual misunderstanding causing the errors. Trap (c) hands over the transposition shortcut as a rule to memorise, exactly the abstract-rule-first approach that bypasses rather than builds the relational understanding needed to make sense of why the rule actually works. Trap (d) avoids the issue rather than resolving it — variable-on-both-sides equations are exactly the format where an operational misreading of '=' causes the most difficulty, so avoiding them leaves the underlying gap untouched.

Question 11

A teacher notices that when asked to write a formula for the total number of legs (L) on t tables, each with 4 legs, a student writes 'L = 4 tables' instead of 'L = 4t'. This error best illustrates which specific misconception?
(a) A sign error in translating the word problem  (b) Treating the variable as a label for an object rather than as a number  (c) Confusion between direct and inverse variation  (d) An error in the distributive property

Accepted Answer

The student's formula treats 't' as if it names the object 'tables' rather than standing for a number (the count of tables) — the well-documented misconception of reading a variable as an object-label rather than a numerical value.

Answer: (b). Trap (a) misnames this as a sign error, but no positive or negative sign is involved in the mistake at all. Trap (c) misapplies a different pedagogy concept — this is a fixed-rate relationship (4 legs per table), correctly a case of direct variation once expressed properly, but the ERROR itself isn't a variation-type confusion, it's about what the letter 't' represents in the first place. Trap (d) misapplies the distributive property, an operation-related identity, to what is really a conceptual misunderstanding of a variable's basic meaning.

Practice — "Mathematics — Algebra"

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