Fractions in Disguise — Practice

Question 1

Express 87.5% as a fraction in simplest form.

Accepted Answer

✦ Answer: 87.5% = 87.5/100 = 875/1000 = 7/8 (after dividing by HCF = 125).

Question 2

Convert 0.5̄ (repeating) to fraction.

Accepted Answer

✦ Answer: Let x = 0.555... Then 10x = 5.555..., so 9x = 5, giving x = 5/9.

Question 3

Convert 0.272̄ (where 272 repeats) to a fraction in simplest form.

Accepted Answer

Step 1 — Set up.
  Let x = 0.272272272...

Step 2 — Multiply by 1000 (since 3 digits repeat).
  1000x = 272.272272...

Step 3 — Subtract.
  1000x − x = 272.272... − 0.272...
  999x = 272
  x = 272/999

Step 4 — Simplify.
  HCF(272, 999) = 1 (272 = 16×17; 999 = 27×37 — no common factor)
  Wait, let me verify: 272 = 16×17; 999 = 3³ × 37. No common factors.
  So x = 272/999 is in lowest form.

✦ Answer: 0.272̄ = 272/999.

Question 4

Simplify: (x² − 4) / (x² + x − 6)

Accepted Answer

Step 1 — Factorise numerator.
  x² − 4 = (x + 2)(x − 2) [difference of squares]

Step 2 — Factorise denominator.
  x² + x − 6: find p, q with p+q = 1, p×q = −6.
  Try 3 and −2: 3 + (−2) = 1 ✓; 3 × (−2) = −6 ✓.
  So x² + x − 6 = (x + 3)(x − 2).

Step 3 — Substitute and cancel.
  (x+2)(x−2) / ((x+3)(x−2))
  Cancel (x−2): = (x+2)/(x+3)

Step 4 — State restriction.
  Result valid for x ≠ 2 (which made denominator zero before cancellation).

✦ Answer: (x² − 4)/(x² + x − 6) = (x+2)/(x+3), for x ≠ 2.

Question 5

Simplify: 1/(x−1) + 1/(x+1) − 2x/(x²−1)

Accepted Answer

Step 1 — Observe common denominator.
  x² − 1 = (x − 1)(x + 1) (difference of squares)
  LCD = (x − 1)(x + 1) = x² − 1

Step 2 — Convert each fraction.
  1/(x−1) = (x+1)/(x²−1)
  1/(x+1) = (x−1)/(x²−1)
  2x/(x²−1) is already over LCD

Step 3 — Combine.
  = [(x+1) + (x−1) − 2x] / (x² − 1)
  = (x + 1 + x − 1 − 2x) / (x² − 1)
  = (2x − 2x) / (x² − 1)
  = 0 / (x² − 1)
  = 0

Step 4 — Verify.
  The three fractions cancel exactly. The expression evaluates to 0 for all x ≠ ±1.

Step 5 — State restriction.
  Result holds for x ≠ 1 and x ≠ −1.

✦ Answer: The expression equals 0 (for x ≠ ±1).

Practice — Fractions in Disguise

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