A Story of Numbers — Practice

Question 1

Express 24/(-36) in standard form.

Accepted Answer

✦ Answer: Make denominator positive: -24/36. HCF(24,36) = 12. Divide both: -24/12 = -2; 36/12 = 3. Standard form: -2/3.

Question 2

Compute: (3/5) + (2/5)

Accepted Answer

✦ Answer: Same denominator. Add numerators: (3+2)/5 = 5/5 = 1.

Question 3

Verify the distributive property for: (2/3) × (1/2 − 3/4).

Accepted Answer

Step 1 — LHS computation.
  Bracket first: 1/2 − 3/4. LCM = 4. So 2/4 − 3/4 = −1/4.
  Now multiply: (2/3) × (−1/4) = −2/12 = −1/6.

Step 2 — RHS computation (distributing).
  (2/3) × (1/2) − (2/3) × (3/4)
  = 2/6 − 6/12
  = 1/3 − 1/2 (after simplification)
  LCM = 6: 2/6 − 3/6 = −1/6.

Step 3 — Compare.
  LHS = −1/6 ✓
  RHS = −1/6 ✓
  EQUAL — distributive property verified.

✦ Answer: Both LHS and RHS equal −1/6. The distributive property a × (b − c) = a × b − a × c holds for rationals.

Question 4

Find three rational numbers between 1/3 and 1/2.

Accepted Answer

Step 1 — Convert to common denominator.
  1/3 = 4/12, 1/2 = 6/12.
  Only one number between (5/12), not three. Need larger denominator.

Step 2 — Multiply numerator and denominator by 4.
  1/3 = 16/48, 1/2 = 24/48.
  Now MANY rationals between.

Step 3 — Pick three.
  17/48, 19/48, 21/48 — any three work.

Step 4 — Verify each is between 1/3 and 1/2.
  16/48 = 1/3 ✓ (boundary)
  17/48: 17/48 > 16/48 ✓ and 17/48 < 24/48 ✓ → between.
  Similarly for 19/48 and 21/48.

✦ Answer: Three rationals between 1/3 and 1/2 are 17/48, 19/48, and 21/48. (Many other answers possible — between any two rationals there are infinitely many others.)

Question 5

A train travels 3/4 of a journey in 2 1/2 hours. At the same speed, how long would the full journey take? Express the time as a rational number in standard form.

Accepted Answer

Step 1 — Set up the problem.
  Distance covered: (3/4) of total journey
  Time taken: 2 1/2 hours = 5/2 hours

Step 2 — Find time per unit fraction.
  Time for 3/4 of journey = 5/2 hours
  Time for 1/4 of journey = (5/2) ÷ 3 = 5/6 hours

Step 3 — Find time for full journey.
  Total = 4 × (5/6) = 20/6 hours
  Simplify: 20/6 = 10/3 hours

Step 4 — Verify.
  Check: 3/4 of 10/3 = (3 × 10)/(4 × 3) = 30/12 = 5/2 ✓

Step 5 — Express in mixed number form.
  10/3 = 3 1/3 hours = 3 hours 20 minutes

✦ Answer: The full journey takes 10/3 hours = 3 1/3 hours = 3 hours 20 minutes. In standard rational form: 10/3.

Practice — A Story of Numbers

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