Question 1

Add simple decimals and express the sum cleanly: Evaluate (0.6 + 0.7 + 0.8 + 0.3) and select an equivalent exact form.

Accepted Answer

Introduction / Context: This exercise reinforces decimal addition and equivalence between decimals and fractions in simplest form. Given Data / Assumptions: Add 0.6, 0.7, 0.8, and 0.3. Provide an exact expression of the result, possibly as a reduced fraction. Concept / Approach: Group tenths to make whole numbers where possible, then convert the decimal total to a fraction by recognizing tenths equivalences. Step-by-Step Solution: 0.6 + 0.4 = 1.0 (we can pair 0.6 with part of 0.7 and 0.3)Remaining after using 0.4 from (0.7 + 0.3) is 0.6Total = 1.0 + 0.6 + 0.8 = 2.42.4 as a fraction = 24/10 = 12/5 Verification / Alternative check: Add directly: 0.6 + 0.7 = 1.3; 1.3 + 0.8 = 2.1; 2.1 + 0.3 = 2.4. Then express 2.4 = 12/5. Why Other Options Are Wrong: 23/10 = 2.3 and 233/100 = 2.33 are close but not equal to 2.4. 2.33 is a decimal approximation not matching. 22/3 ≈ 7.33 is far off. Common Pitfalls: Misplacing a decimal or assuming 0.6 + 0.7 = 1.2 (incorrect); the correct is 1.3. Always align tenths correctly. Final Answer: 12/5

Question 2

Parameter ratio from a linear relation: If 1.5 * a = 0.04 * b, find the value of (b − a) / (b + a).

Accepted Answer

Introduction / Context: This question tests using a given proportional relation between two variables to compute a derived ratio. Recognizing and simplifying ratios prevents unnecessary algebra. Given Data / Assumptions: Relation: 1.5 * a = 0.04 * b. We need (b − a) / (b + a). Concept / Approach: Compute r = b/a from the given equation, then express the required ratio in terms of r only: (b − a)/(b + a) = (r − 1)/(r + 1). Step-by-Step Solution: From 1.5a = 0.04b ⇒ b/a = 1.5 / 0.04 = 37.5Let r = 37.5. Then (b − a)/(b + a) = (r − 1)/(r + 1)Compute (37.5 − 1)/(37.5 + 1) = 36.5 / 38.5Multiply numerator and denominator by 2 to clear halves: 73 / 77 Verification / Alternative check: Pick a convenient a (e.g., a = 2). Then b = 37.5 * 2 = 75. Compute (75 − 2)/(75 + 2) = 73/77, confirming the general result. Why Other Options Are Wrong: 36/38 is a rounded version (reduces to 18/19) and not exact. 77/73, 73/70, and 73/75 invert or perturb the ratio and do not satisfy the derived expression. Common Pitfalls: Computing b/a incorrectly (e.g., inverting 1.5/0.04), or forgetting to transform (r − 1)/(r + 1) exactly, leading to rounding errors. Final Answer: 73/77

Question 3

Resolve a radical equation and identify the invariant ratio: If √0.05 * 0.5 * a = 0.5 * 0.05 * √b, find the value of a^2 / b.

Accepted Answer

Introduction / Context: The original statement asked for a/b, which cannot be uniquely determined from the given relation. Using the recovery-first policy, we minimally refine the target to a^2 / b, an invariant ratio fully determined by the equation. This preserves the intent and yields a solvable, well-posed task. Given Data / Assumptions: Equation: √0.05 * 0.5 * a = 0.5 * 0.05 * √b. All quantities are real and positive for the square roots to be real. Goal: compute a^2 / b. Concept / Approach: First cancel the common factor 0.5 from both sides. Then isolate a in terms of √b, and square to obtain a relation between a^2 and b. Finally, form the requested ratio. Step-by-Step Solution: Cancel 0.5: √0.05 * a = 0.05 * √bSolve for a: a = (0.05 / √0.05) * √b = √0.05 * √b = √(0.05 * b)Square both sides: a^2 = 0.05 * bTherefore, a^2 / b = 0.05 Verification / Alternative check: Pick b = 100 for convenience: a^2 = 0.05 * 100 = 5 ⇒ a = √5. Substituting into the original equation satisfies the equality and yields a^2 / b = 5 / 100 = 0.05, confirming the result. Why Other Options Are Wrong: 0.0025 and 0.025 are smaller by factors of 20 or 2. 0.25 is too large. The derived identity fixes the ratio exactly at 0.05. Common Pitfalls: Attempting to compute a/b directly (underdetermined), or mishandling the division 0.05 / √0.05 instead of recognizing it equals √0.05. Final Answer: 0.05

Question 4

Handle nested decimals and a complex division: Compute (.23 − .023) / (.0023 / 23) and choose the exact value.

Accepted Answer

Introduction / Context: This problem combines decimal subtraction with a division by a small decimal ratio, testing careful manipulation of place values and reciprocals. Given Data / Assumptions: Numerator: 0.23 − 0.023. Denominator: 0.0023 / 23. Perform exact arithmetic (no rounding required). Concept / Approach: First simplify the numerator. Then recognize that dividing by a small number is equivalent to multiplying by its reciprocal; compute the denominator exactly to avoid compounding decimal errors. Step-by-Step Solution: Numerator: 0.23 − 0.023 = 0.207Denominator: 0.0023 / 23 = 0.0001 (since 23 × 0.0001 = 0.0023)Overall value = 0.207 / 0.0001 = 2070 Verification / Alternative check: Rewrite the compound fraction: (0.23 − 0.023) * (23 / 0.0023) = 0.207 * 10000 = 2070, the same result. Why Other Options Are Wrong: 0.207 and 0.0207 ignore the denominator’s effect. 207 under-multiplies by a factor of 10. 20.7 is still short by two orders of magnitude. Common Pitfalls: Miscomputing 0.0023 / 23 or moving the decimal point incorrectly when dividing by a ten-thousandth. Final Answer: 2070

Question 5

Use an identity to simplify a ratio of sums of squares: Evaluate [(3.537 − 0.948)^2 + (3.537 + 0.948)^2] / [(3.537)^2 + (0.948)^2].

Accepted Answer

Introduction / Context: This expression is designed for application of a standard algebraic identity. Recognizing structure avoids arduous arithmetic with decimals. Given Data / Assumptions: a = 3.537, b = 0.948. Expression: [(a − b)^2 + (a + b)^2] / (a^2 + b^2). Concept / Approach: Identity: (a − b)^2 + (a + b)^2 = 2a^2 + 2b^2 = 2(a^2 + b^2). Substitute this into the expression to simplify immediately. Step-by-Step Solution: Numerator = (a − b)^2 + (a + b)^2 = 2(a^2 + b^2)Denominator = (a^2 + b^2)Ratio = [2(a^2 + b^2)] / (a^2 + b^2) = 2 Verification / Alternative check: Compute a^2 + b^2 numerically if desired; the factor of 2 cancels regardless of specific values, confirming the identity-based simplification. Why Other Options Are Wrong: 4 and 4.485 overstate the result; 2.589 is arbitrary. 1 would occur if the numerator equaled the denominator, which is not the case here. Common Pitfalls: Expanding each square separately and then adding may invite arithmetic errors. Spot the identity first to save time. Final Answer: 2

Question 6

Factor and simplify before computing: Evaluate (6.5 × 4.7 + 6.5 × 5.3) / (1.3 × 7.9 − 1.3 × 6.9).

Accepted Answer

Introduction / Context: Recognizing common factors can drastically simplify decimal arithmetic. This problem is constructed to reward factoring before multiplying. Given Data / Assumptions: Numerator: 6.5 × 4.7 + 6.5 × 5.3. Denominator: 1.3 × 7.9 − 1.3 × 6.9. Concept / Approach: Factor common terms: a*b + a*c = a(b + c); a*b − a*c = a(b − c). This avoids unnecessary decimal products and leads to a clean ratio. Step-by-Step Solution: Numerator = 6.5(4.7 + 5.3) = 6.5 × 10 = 65Denominator = 1.3(7.9 − 6.9) = 1.3 × 1 = 1.3Value = 65 / 1.3 = 50 Verification / Alternative check: Compute directly as a decimal: 4.7 + 5.3 = 10 gives 65 on top; 7.9 − 6.9 = 1 gives 1.3 below; then 65 ÷ 1.3 = 650 ÷ 13 = 50 exactly. Why Other Options Are Wrong: 39 and 3.9 confuse multiplication with division by 1.3. 34.45 is a non-factored calculator mishit. 5 is off by a factor of 10. Common Pitfalls: Multiplying decimals first and rounding prematurely, which can lead to compounded errors; always look to factor common terms. Final Answer: 50

Question 7

Choose the greatest fraction among the following options Compare 2/5, 5/6, 11/12, and 7/8 and identify which fraction has the largest value.

Accepted Answer

Introduction / Context: Comparing fractions is a frequent aptitude skill. Here you must determine which of 2/5, 5/6, 11/12, and 7/8 is the greatest without relying on a calculator. We use quick estimation and cross-multiplication to rank them correctly. Given Data / Assumptions: Candidates: 2/5, 5/6, 11/12, 7/8. All are positive proper fractions. Goal: find the maximum value. Concept / Approach: Two reliable approaches exist: (1) convert to approximate decimals, or (2) compare in pairs using cross-multiplication (this avoids computing decimals). For a/b and c/d, compare a*d with c*b. The larger cross-product corresponds to the larger fraction. Step-by-Step Solution: Estimate: 2/5 = 0.4; 5/6 ≈ 0.8333; 7/8 = 0.875; 11/12 ≈ 0.9167.Ordering by size from largest to smallest: 11/12, 7/8, 5/6, 2/5.Therefore, the greatest fraction is 11/12. Verification / Alternative check: Compare 11/12 with 7/8 using cross-multiplication: 11*8 = 88 vs. 7*12 = 84. Since 88 > 84, 11/12 > 7/8. Similarly, 7/8 > 5/6 because 7*6 = 42 vs. 8*5 = 40. Why Other Options Are Wrong: 7/8 (0.875) and 5/6 (≈0.8333) are both less than 11/12 (≈0.9167); 2/5 equals 0.4, far smaller. A distractor like 13/16 (0.8125) is also less than 11/12. Common Pitfalls: Comparing only numerators or denominators; assuming the fraction with the largest denominator is smallest, which is not always true across sets; rounding too coarsely. Final Answer: 11/12

Question 8

Evaluate the identity-style expression accurately Compute the exact value of (.6)^3 + (.4)^3 + 3 * (.6) * (.4) * (.6 + .4).

Accepted Answer

Introduction / Context: The original text was ambiguous, but by applying the Recovery-First policy we interpret it as a classic algebraic identity: a^3 + b^3 + 3ab(a + b) with a = 0.6 and b = 0.4. Recognizing identities lets you compute quickly and exactly. Given Data / Assumptions: Expression: (.6)^3 + (.4)^3 + 3 * (.6) * (.4) * (.6 + .4). a = 0.6, b = 0.4. We use the binomial cube identity. Concept / Approach: The identity is a^3 + b^3 + 3ab(a + b) = (a + b)^3. This collapses the sum to a single cube, avoiding term-by-term arithmetic and rounding errors. Step-by-Step Solution: Identify a = 0.6, b = 0.4.Compute a + b = 0.6 + 0.4 = 1.0.Apply identity: a^3 + b^3 + 3ab(a + b) = (a + b)^3.Therefore value = (1.0)^3 = 1. Verification / Alternative check: Compute numerically: 0.6^3 = 0.216; 0.4^3 = 0.064; 3 * 0.6 * 0.4 * 1.0 = 0.72. Sum = 0.216 + 0.064 + 0.72 = 1.0 exactly. Why Other Options Are Wrong: Values like 21.736 or 2.1736 are orders of magnitude off; 0.21736 confuses partial multiplication; 0.733824 equals (0.6^3 * 0.4^3) + 0.72 (a misapplied product), not the intended identity. Common Pitfalls: Misreading the expression as chained products; forgetting the identity or misapplying coefficients; rounding early. Final Answer: 1

Question 9

Use the (a − b)^3 pattern to simplify Evaluate (.58)^3 − (.42)^3 − 3 * (.58) * (.42) * (.58 − .42) exactly.

Accepted Answer

Introduction / Context: This expression is designed for the binomial cube identity. Recognizing the structure a^3 − b^3 − 3ab(a − b) allows immediate simplification to (a − b)^3, which is simple to compute. Given Data / Assumptions: a = 0.58, b = 0.42. Expression: a^3 − b^3 − 3ab(a − b). a − b = 0.16. Concept / Approach: Identity: (a − b)^3 = a^3 − b^3 − 3ab(a − b). Hence the entire expression equals (a − b)^3. This avoids tedious cubic computations. Step-by-Step Solution: Compute a − b = 0.58 − 0.42 = 0.16.By identity, value = (a − b)^3 = (0.16)^3.0.16^2 = 0.0256; multiply by 0.16: 0.0256 * 0.16 = 0.004096. Verification / Alternative check: Expanding (0.16)^3 directly or evaluating a^3, b^3, and 3ab(a − b) numerically will agree with 0.004096. Why Other Options Are Wrong: 1.3976 and 1 are far too large; 0.16 is (a − b), not its cube; 0.04096 is off by a factor of 10. Common Pitfalls: Forgetting the exact identity; cubing only the difference once; decimal place errors when cubing small decimals. Final Answer: 0.004096

Question 10

Solve for the original fraction from two conditions If 3 is added to the denominator, the fraction becomes 1/3; if 4 is added to the numerator, it becomes 3/4. Find the original fraction.

Accepted Answer

Introduction / Context: Classic fraction-reconstruction problems translate words into equations in x/y, then solve a simple linear system. Here, two alterations to numerator or denominator yield target fractions 1/3 and 3/4. Given Data / Assumptions: Original fraction = x/y, with positive integers x, y. x/(y + 3) = 1/3 when 3 is added to denominator. (x + 4)/y = 3/4 when 4 is added to numerator. Concept / Approach: Convert to linear equations and solve simultaneously. Keep arithmetic exact to avoid rounding errors. Step-by-Step Solution: From x/(y + 3) = 1/3 ⇒ 3x = y + 3 ⇒ y = 3x − 3.From (x + 4)/y = 3/4 ⇒ 4x + 16 = 3y.Substitute y: 4x + 16 = 3(3x − 3) = 9x − 9.Rearrange: 16 + 9 = 9x − 4x ⇒ 25 = 5x ⇒ x = 5.Then y = 3*5 − 3 = 12. Fraction = 5/12. Verification / Alternative check: Check 5/(12 + 3) = 5/15 = 1/3 and (5 + 4)/12 = 9/12 = 3/4. Both conditions hold. Why Other Options Are Wrong: 4/9, 3/20, 7/24, 2/9 do not satisfy both transformation rules when tested. Common Pitfalls: Mixing which change applies to numerator vs. denominator; arithmetic slips when substituting. Final Answer: 5/12

Question 11

Determine the original fraction from two transformed forms (x + 2)/(y + 1) = 5/8 and (x + 3)/(y + 1) = 3/4. Find the original fraction x/y.

Accepted Answer

Introduction / Context: This problem provides two versions of the same fraction after specific changes to numerator and denominator. By translating to equations in x and y, we can solve the resulting linear system uniquely. Given Data / Assumptions: (x + 2)/(y + 1) = 5/8. (x + 3)/(y + 1) = 3/4. Original fraction is x/y in lowest terms. Concept / Approach: Cross-multiply each equation to create two linear relations. Solve simultaneously to obtain x and y. Step-by-Step Solution: From (x + 2)/(y + 1) = 5/8 ⇒ 8x + 16 = 5y + 5 ⇒ 8x − 5y = −11.From (x + 3)/(y + 1) = 3/4 ⇒ 4x + 12 = 3y + 3 ⇒ 4x − 3y = −9.Multiply the second by 2: 8x − 6y = −18.Subtract: (8x − 5y) − (8x − 6y) = −11 − (−18) ⇒ y = 7.Use 4x − 3y = −9 ⇒ 4x − 21 = −9 ⇒ 4x = 12 ⇒ x = 3.Original fraction = 3/7. Verification / Alternative check: Check: (3 + 2)/(7 + 1) = 5/8 and (3 + 3)/(7 + 1) = 6/8 = 3/4. Both conditions match. Why Other Options Are Wrong: 2/7 and 4/7 fail at least one of the conditions; “Data inadequate” is incorrect since the system has a unique solution. Common Pitfalls: Sign errors when moving terms; forgetting to increase the denominator by the same amount in both equations as stated. Final Answer: 3/7

Question 12

Find the original fraction using two changes Given (x + 2)/(y + 3) = 7/9 and (x − 1)/(y − 1) = 4/5, determine x/y.

Accepted Answer

Introduction / Context: Two independent transformations of a fraction are given. Solving the resulting pair of linear equations in x and y yields the original fraction. Given Data / Assumptions: (x + 2)/(y + 3) = 7/9. (x − 1)/(y − 1) = 4/5. x, y are positive integers representing the original fraction x/y. Concept / Approach: Cross-multiply both equations to create linear relations and solve simultaneously using elimination or substitution. Step-by-Step Solution: From (x + 2)/(y + 3) = 7/9 ⇒ 9x + 18 = 7y + 21 ⇒ 9x − 7y = 3.From (x − 1)/(y − 1) = 4/5 ⇒ 5x − 5 = 4y − 4 ⇒ 5x − 4y = 1.Eliminate y: multiply the second equation by 7 ⇒ 35x − 28y = 7.Multiply the first by 4 ⇒ 36x − 28y = 12.Subtract: (36x − 28y) − (35x − 28y) = 12 − 7 ⇒ x = 5.Substitute: 5x − 4y = 1 ⇒ 25 − 4y = 1 ⇒ y = 6.Original fraction = 5/6. Verification / Alternative check: Check both: (5 + 2)/(6 + 3) = 7/9 and (5 − 1)/(6 − 1) = 4/5. Works exactly. Why Other Options Are Wrong: 13/16, 9/11, and 17/21 do not satisfy both equations; 7/9 is a target fraction after a change, not the original. Common Pitfalls: Arithmetic slips when aligning coefficients; mixing which equation corresponds to which adjustment. Final Answer: 5/6

Question 13

Recover the original fraction from two scenarios If (x + 1)/(y + 2) = 2/3 and (x + 5)/(y + 1) = 5/4, find the original fraction x/y.

Accepted Answer

Introduction / Context: Two altered versions of a fraction are provided. Interpreting these as equations allows solving for x and y systematically. Given Data / Assumptions: (x + 1)/(y + 2) = 2/3. (x + 5)/(y + 1) = 5/4. x/y is the unknown original fraction. Concept / Approach: Cross-multiply to obtain linear equations, then eliminate one variable. Keep steps exact and consistent with the stated adjustments. Step-by-Step Solution: From (x + 1)/(y + 2) = 2/3 ⇒ 3x + 3 = 2y + 4 ⇒ 3x − 2y = 1.From (x + 5)/(y + 1) = 5/4 ⇒ 4x + 20 = 5y + 5 ⇒ 4x − 5y = −15.Multiply first by 5: 15x − 10y = 5.Multiply second by 2: 8x − 10y = −30.Subtract equations: 7x = 35 ⇒ x = 5.Use 3x − 2y = 1 ⇒ 15 − 2y = 1 ⇒ 2y = 14 ⇒ y = 7.Original fraction = 5/7. Verification / Alternative check: Check: (5 + 1)/(7 + 2) = 6/9 = 2/3 and (5 + 5)/(7 + 1) = 10/8 = 5/4. Both match. Why Other Options Are Wrong: 3/7 and 5/8 do not satisfy both conditions; 6/7 is too large; 7/5 is improper and incorrect. Common Pitfalls: Misreading the increments to numerator/denominator; arithmetic mistakes in elimination. Final Answer: 5/7

Question 14

Simplify powers using prime-factorization rules Compute the exact value of (9^2 * 18^4) / 3^16.

Accepted Answer

Introduction / Context: Exponent problems with composite bases are best handled by prime factorization. Converting all terms to powers of the same primes allows straightforward cancellation. Given Data / Assumptions: Expression: (9^2 * 18^4) / 3^16. 3 is the key prime in all bases. Concept / Approach: Rewrite 9 and 18 in terms of 3 (and 2 for 18). Use a^m * a^n = a^(m+n) and a^m / a^n = a^(m−n). Keep factors of 2 separate. Step-by-Step Solution: 9 = 3^2 ⇒ 9^2 = (3^2)^2 = 3^4.18 = 2 * 3^2 ⇒ 18^4 = 2^4 * (3^2)^4 = 2^4 * 3^8.Numerator: 3^4 * (2^4 * 3^8) = 2^4 * 3^(4+8) = 2^4 * 3^12.Divide by 3^16 ⇒ 2^4 * 3^(12−16) = 2^4 * 3^(−4) = 16 / 81. Verification / Alternative check: Compute via exponent subtraction directly or evaluate numerically to confirm ≈ 0.1975… equals 16/81. Why Other Options Are Wrong: 2/3, 4/9, 32/243, 8/27 are different reductions that do not match the exact exponent arithmetic here. Common Pitfalls: Forgetting 18 includes 2; mis-adding exponents; dividing powers incorrectly. Final Answer: 16/81

Question 15

Successive fraction-of-remaining spending with a rupee balance A man spends 1/7 of salary on food, half of the remaining on clothing, and 1/3 of the remaining on entertainment, and is left with Rs 600. How much did he spend on entertainment?

Accepted Answer

Introduction / Context: This is a classic “fraction of the remainder” problem. Tracking successive expenditures as fractions of what remains at each stage and relating the final balance to the initial salary yields the entertainment amount. Given Data / Assumptions: Salary = S (rupees). Food: spends 1/7 of S. Clothing: spends 1/2 of the remaining after food. Entertainment: spends 1/3 of the remaining after clothing. Final amount left = Rs 600. Concept / Approach: Work forward with remainders or backward from the final balance using inverse operations. Either way, express each stage in terms of S and solve. Step-by-Step Solution: After food: remaining = S − (1/7)S = (6/7)S.Clothing: spend (1/2) of remaining ⇒ (1/2)*(6/7)S = (3/7)S, leaving (6/7)S − (3/7)S = (3/7)S.Entertainment: spend (1/3) of remaining ⇒ (1/3)*(3/7)S = (1/7)S, leaving (3/7)S − (1/7)S = (2/7)S.Given leftover (2/7)S = 600 ⇒ S = 600 * (7/2) = 2100.Entertainment spend = (1/7)S = 2100/7 = Rs 300. Verification / Alternative check: Forward check: Food 300, remain 1800; Clothing 900, remain 900; Entertainment 300, remain 600 — consistent. Why Other Options Are Wrong: Rs 600 is the final balance, not the entertainment spend; Rs 450 and Rs 700 do not match the computed sequence; Rs 200 is inconsistent with the derived salary. Common Pitfalls: Applying fractions to the original salary instead of the remainder; mixing up “spend” vs. “left” amounts; arithmetic slips with sevenths. Final Answer: Rs. 300

Question 16

Find the fraction from a cube-operation description A fraction f is multiplied by itself and then divided by its reciprocal, giving 18 26/27. Determine the fraction.

Accepted Answer

Introduction / Context: This verbal description encodes an algebraic operation on a fraction f. “Multiplied by itself” is f^2, and “divided by its reciprocal” 1/f yields f^2 ÷ (1/f) = f^3. The result is given as a mixed number, which we convert to an improper fraction to solve for f. Given Data / Assumptions: Operation result: 18 26/27. Operation on f: f^2 ÷ (1/f) = f^3. Find f in simplest form. Concept / Approach: Since f^3 equals the given rational, take the rational cube root. Recognize perfect cubes in numerator and denominator for a clean result. Step-by-Step Solution: Convert mixed number: 18 26/27 = (18*27 + 26)/27 = (486 + 26)/27 = 512/27.Set f^3 = 512/27.Note 512 = 8^3 and 27 = 3^3.Therefore f = cube_root(512/27) = 8/3. Verification / Alternative check: Compute (8/3)^3 = 512/27, which equals 18 26/27, confirming the result. Why Other Options Are Wrong: 8/27 is far too small; 11/3, 22/3, 32/3 produce cubes much larger than 512/27. Common Pitfalls: Misinterpreting “divided by the reciprocal” as division by f (which would give f); converting the mixed number incorrectly. Final Answer: 8/3

Question 17

A lamppost is divided into three parts as follows: • The first part is exactly 1/3 of the total length. • The second part measures 3/8 of the first part (that is, a fraction of the first segment). • The third and final part measures 6 m 50 cm (i.e.,…

Accepted Answer

Introduction / Context: In this problem involving fractions of a whole, a lamppost is split into three parts. Two parts are given as fractional portions of the total (or of another part), and the last part is given as a fixed metric length. We are asked to calculate the total length. This tests proportional reasoning, fraction operations, and consistent unit handling (metres and centimetres). Given Data / Assumptions: Total length = L metres (unknown). First part = (1/3) * L. Second part = (3/8) of the first part = (3/8) * (1/3) * L = (1/8) * L. Third part = 6 m 50 cm = 6.5 m. Concept / Approach: Express each segment in terms of L, sum them, and set equal to the whole. Use fraction arithmetic carefully and keep units in metres. Then solve the linear equation for L. Step-by-Step Solution: First part = (1/3)LSecond part = (1/8)LThird part = 6.5Sum of parts = L(1/3)L + (1/8)L + 6.5 = LFind common denominator for the fractions: 1/3 = 8/24, 1/8 = 3/24, so (8/24 + 3/24)L = (11/24)L(11/24)L + 6.5 = L → L - (11/24)L = 6.5(13/24)L = 6.5L = 6.5 * 24 / 13Since 6.5 = 13/2, L = (13/2) * (24/13) = 24/2 = 12 Verification / Alternative check: Compute the actual segments for L = 12: first = 4, second = 1.5, third = 6.5; sum = 4 + 1.5 + 6.5 = 12. Correct. Why Other Options Are Wrong: 15 m, 10 m, 13.5 m, 11.5 m: Each fails when substituted back; the fractional breakdown and fixed 6.5 m segment will not sum consistently to these totals. Common Pitfalls: Confusing “3/8 of the first part” with “3/8 of the whole.” Unit error: forgetting that 6 m 50 cm = 6.5 m. Arithmetic slip when combining 1/3 and 1/8; always convert to a common denominator. Final Answer: 12 m

Question 18

Compute the product below and give the result to two decimal places: 27 × 1.2 × 5.526 × 0.6

Accepted Answer

Introduction / Context: This question assesses accurate multiplication of decimals and integers, along with sensible rounding to a specified precision (two decimal places). Such computations are common in percentage, rate, and measurement problems. Given Data / Assumptions: Expression: 27 × 1.2 × 5.526 × 0.6 All factors are exact as stated. Final answer required to two decimal places. Concept / Approach: Group factors to keep the numbers manageable, multiply progressively, and only round at the end. Avoid intermediate rounding to maintain precision. Step-by-Step Solution: Compute A = 27 × 1.2 = 32.4Compute B = A × 0.6 = 32.4 × 0.6 = 19.44Compute C = B × 5.526 = 19.44 × 5.526 = 107.42544Round C to two decimals → 107.43 Verification / Alternative check: Check reasonableness: 27 × 1.2 ≈ 32.4; 32.4 × 0.6 ≈ 19.44; 19.44 × 5.5 ≈ 106.9, close to 107.43. The order-of-magnitude is consistent. Why Other Options Are Wrong: 121.57, 113.40, 101.74, 95.22: These come from rounding too early, misplacing a decimal, or using an incorrect factor order. Common Pitfalls: Rounding after each multiplication step (accumulates error). Misreading 5.526 as 5.56 or 5.5262. Dropping a factor (e.g., forgetting the 0.6). Final Answer: 107.43

Question 19

A 330 cm metal rod is to be cut into as many equal pieces as possible, each piece being exactly 13.2 cm long. How many pieces can be obtained?

Accepted Answer

Introduction / Context: This is a direct division problem involving lengths with decimals. It checks the ability to divide a total length by the length per piece to find the maximum number of equal pieces with no remainder left over. Given Data / Assumptions: Total rod length = 330 cm. Length of each piece = 13.2 cm. All pieces must be exactly equal in length. Concept / Approach: Number of pieces = total length / piece length. Since both numbers are given in centimetres, no unit conversion is needed. We expect an integer if the division is exact. Step-by-Step Solution: Pieces = 330 / 13.2Remove decimal by scaling: 3300 / 132 = 25Therefore, exactly 25 equal pieces are possible. Verification / Alternative check: Check: 13.2 × 25 = 330.0 cm → perfect match. Why Other Options Are Wrong: 28, 21, 35, 22: These are common guesses when dividing, but none reproduce the exact total length when multiplied back by 13.2 cm. Common Pitfalls: Losing track of the decimal place in 13.2. Rounding 13.2 to 13 or 14, which gives an incorrect count. Final Answer: 25

Question 20

A pencil has three colour segments in order. The first 1/8 of the whole pencil is black; half of the remaining length is white; the rest is blue. If the blue segment measures 3.5 cm, what is the total length of the pencil?

Accepted Answer

Introduction / Context: The question involves successive fractions applied to a whole. After removing the first fraction, another fraction of the remainder is taken, and the leftover is given in centimetres. We must backtrack to the total length. This is a typical fraction-of-a-remainder problem. Given Data / Assumptions: Total length = L cm. Black segment = (1/8)L. Remaining after black = (7/8)L. White segment = (1/2) * (7/8)L = (7/16)L. Blue segment = remaining = (7/8)L - (7/16)L = (7/16)L. Blue segment length is 3.5 cm. Concept / Approach: Set up the expression for the blue segment in terms of L and equate it to the given 3.5 cm to solve for L. Keep all calculations in centimetres for consistency. Step-by-Step Solution: Blue = (7/16)L = 3.5L = 3.5 * (16/7)Since 3.5 = 7/2, L = (7/2) * (16/7) = 16/2 = 8 Verification / Alternative check: Compute segments: black = 1/8 of 8 = 1; remaining = 7; white = half of 7 = 3.5; blue = 7 - 3.5 = 3.5. Totals add to 8 cm, matching L. Why Other Options Are Wrong: 6, 7, 9, 10 cm: Substituting these values fails to keep the final blue length equal to 3.5 cm given the fractional splits. Common Pitfalls: Taking half of the original instead of half of the remainder. Arithmetic errors when subtracting fractional parts. Final Answer: 8 cm

Decimal Fraction Questions