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Simplification Aptitude Quiz

Question 1

Simplify: (957 + 932)² - 4 x 957 x 932

Correct Answer: 625

Solution Explanation: This expression matches the algebraic identity $(a+b)² - 4ab$.

  • We know that $(a+b)² - 4ab = (a² + 2ab + b²) - 4ab = a² - 2ab + b² = (a-b)²$.
  • Here, let a = 957 and b = 932.
  • The expression becomes $(957 - 932)²$.
  • $(25)² = \textbf{625}$.

Question 2

Simplify: 9 / [5 + 7 / {9+9 / (9 + 9 / 4)}]

Correct Answer: 63/40

Solution Explanation: We solve this by working from the innermost bracket outwards, following the BODMAS rule.

  1. Innermost bracket: $9 + 9/4 = 36/4 + 9/4 = 45/4$.
  2. Next level (division): $9 / (45/4) = 9 \times 4/45 = 36/45 = 4/5$.
  3. Curly braces: $9 + 4/5 = 45/5 + 4/5 = 49/5$.
  4. Next level (division): $7 / (49/5) = 7 \times 5/49 = 35/49 = 5/7$.
  5. Square brackets: $5 + 5/7 = 35/7 + 5/7 = 40/7$.
  6. Final division: $9 / (40/7) = 9 \times 7/40 = \textbf{63/40}$.

Question 3

Simplify: 983² - 17²

Correct Answer: 966,000

Solution Explanation: This expression is in the form of the algebraic identity $a² - b²$.

  • We know that $a² - b² = (a - b)(a + b)$.
  • Here, let a = 983 and b = 17.
  • The expression becomes $(983 - 17)(983 + 17)$.
  • $(966)(1000) = \textbf{966,000}$.

Question 4

What is the value of 11 / [ 1 + 1 / { 1 + 1 / ( 1 + 1/ 3 ) } ]?

Correct Answer: 7

Solution Explanation: This is a continued fraction. We solve it from the bottom up.

  1. Innermost part: $1 + 1/3 = 4/3$.
  2. Next level: $1 / (4/3) = 3/4$.
  3. Next level: $1 + 3/4 = 7/4$.
  4. Next level: $1 / (7/4) = 4/7$.
  5. Final denominator: $1 + 4/7 = 11/7$.
  6. Final expression: $11 / (11/7) = 11 \times 7/11 = \textbf{7}$.

Question 5

Simplify: 24 - [52 - {4 + 10 x (-3)}]

Correct Answer: -54

Solution Explanation: We solve using the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction).

  1. Innermost braces {}: First, multiply 10 x (-3) = -30. Then add 4 + (-30) = -26.
  2. Square brackets []: 52 - {-26} becomes 52 + 26 = 78.
  3. Final calculation: 24 - 78 = -54.

Question 6

Simplify: ((82 x 82) - ( 18 x 18))¹/²

Correct Answer: 80

Solution Explanation: The expression can be written as (82² - 18²)¹/² or √(82² - 18²).

  • We use the identity $a² - b² = (a - b)(a + b)$.
  • √((82 - 18)(82 + 18))
  • √((64)(100))
  • √6400 = 80.

Question 7

What is the value of ( 1 1/4 ÷ 2 3/5 ) ÷ 1 7/13?

Correct Answer: 5/16

Solution Explanation:

  1. Convert all mixed fractions to improper fractions:
    • $1\ 1/4 = 5/4$
    • $2\ 3/5 = 13/5$
    • $1\ 7/13 = 20/13$
  2. Solve the part in the parenthesis first:
    • $5/4 ÷ 13/5 = 5/4 \times 5/13 = 25/52$.
  3. Perform the final division:
    • $25/52 ÷ 20/13 = 25/52 \times 13/20$.
    • Simplify: $\frac{25}{20} \times \frac{13}{52} = \frac{5}{4} \times \frac{1}{4} = \textbf{5/16}$.

Question 8

Simplify p³ + 27.

Correct Answer: (p + 3)(p² - 3p + 9)

Solution Explanation: This is in the form of the sum of cubes identity: $a³ + b³ = (a + b)(a² - ab + b²)$.

  • Here, a = p and b = 3 (since $3³ = 27$).
  • Substituting into the formula gives: (p + 3)(p² - (p)(3) + 3²).
  • This simplifies to (p + 3)(p² - 3p + 9).

Question 9

If 2945 / 19 - 11 x 5 = 29 + a, find the value of a.

Correct Answer: 71

Solution Explanation: First, simplify the left-hand side (LHS) using BODMAS.

  1. Division: 2945 / 19 = 155.
  2. Multiplication: 11 x 5 = 55.
  3. Subtraction: 155 - 55 = 100.
  4. So the equation is 100 = 29 + a.
  5. Solve for a: a = 100 - 29 = 71.

Question 10

What is the value of (4³ + 4) / [5² - (7² - 41)]?

Correct Answer: 4

Solution Explanation:

  1. Simplify the numerator: $4³ + 4 = 64 + 4 = 68$.
  2. Simplify the denominator:
    • Innermost bracket: $7² - 41 = 49 - 41 = 8$.
    • Main denominator: $5² - 8 = 25 - 8 = 17$.
  3. Perform the final division: $68 / 17 = \textbf{4}$.

Question 11

Simplify: (87 x 87 + 61 x 61 - 2 x 87 x 61)

Correct Answer: 676

Solution Explanation: This expression matches the algebraic identity $a² + b² - 2ab$.

  • We know that $a² + b² - 2ab = (a - b)²$.
  • Here, let a = 87 and b = 61.
  • The expression becomes $(87 - 61)²$.
  • $(26)² = \textbf{676}$.

Question 12

Simplify: 3 1/3 ÷ 6 3/7 of 1 1/4

Correct Answer: 56/135

Solution Explanation: According to BODMAS, 'of' is calculated before 'division'.

  1. Convert to improper fractions: $10/3 ÷ 45/7$ of $5/4$.
  2. Calculate the 'of' part first: $45/7 \times 5/4 = 225/28$.
  3. Now the expression is: $10/3 ÷ 225/28$.
  4. Perform the division: $10/3 \times 28/225$.
  5. Simplify: $\frac{10}{225} \times \frac{28}{3} = \frac{2}{45} \times \frac{28}{3} = \textbf{56/135}$.

Question 13

If x/y = 6/5, find the value of (x² + y²) / (x² - y²).

Correct Answer: 61/11

Solution Explanation:

  1. Divide the numerator and the denominator of the expression by y²:
    • $\frac{(x²/y²) + (y²/y²)}{(x²/y²) - (y²/y²)} = \frac{(x/y)² + 1}{(x/y)² - 1}$
  2. Substitute the given value x/y = 6/5 into the new expression:
    • $\frac{(6/5)² + 1}{(6/5)² - 1} = \frac{(36/25) + 1}{(36/25) - 1}$
  3. Simplify the numerator and denominator:
    • Numerator: $36/25 + 25/25 = 61/25$.
    • Denominator: $36/25 - 25/25 = 11/25$.
  4. Perform the final division: $(61/25) / (11/25) = 61/25 \times 25/11 = \textbf{61/11}$.

Question 14

Simplify: (10² + 11² + 12²) / (13² - 160)

Correct Answer: 365/9

Solution Explanation:

  1. Simplify the numerator: $100 + 121 + 144 = 365$.
  2. Simplify the denominator: $169 - 160 = 9$.
  3. The final simplified fraction is 365/9.

Question 15

Find the value of √72 + √72 + √72 + ... up to infinity.

Correct Answer: 9

Solution Explanation: This is an infinite nested radical problem.

  1. Let x = √72 + √72 + √72 + ...
  2. Since the expression is infinite, we can write: x = √(72 + x).
  3. Square both sides to remove the radical: x² = 72 + x.
  4. Rearrange into a quadratic equation: x² - x - 72 = 0.
  5. Factor the quadratic equation. We need two numbers that multiply to -72 and add to -1. These numbers are -9 and +8.
    • (x - 9)(x + 8) = 0.
  6. The possible values for x are 9 and -8. Since the square root of a positive number cannot be negative, we discard -8.
  7. Therefore, x = 9.