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MathidiaMathematics Academy
Exact number and rational reasoning
Lesson 15 of 75
Review

Rational-number problems

Objective. Solve multi-step problems requiring exact rational arithmetic.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Rational-number problems require combining several exact fraction or ratio operations across multiple steps without converting to decimals, which would lose precision. The method is to plan the order of operations carefully, keeping every intermediate result as a fraction, and simplifying only at sensible points rather than after every single step. For example, finding 3/4 of a quantity, then reducing that result by 1/5, needs the fractions multiplied and subtracted exactly, such as 3/4 × (4/5) if asked for a remaining fraction. These problems often appear as worded contexts involving money, lengths or probability where an exact rational answer is required rather than an approximation.

Worked example

A tank is 2/3 full. After 1/4 of the water is removed, what fraction of the tank is left full, as an exact rational number?

2/31/4
Fraction bar showing 2 out of 3 parts shaded
  1. 1The amount removed is 1/4 of 2/3: 1/4 × 2/3 = 2/12 = 1/6.
  2. 2Subtract this removed amount from the original fraction: 2/3 − 1/6.
  3. 3Convert to a common denominator: 2/3 = 4/6, so 4/6 − 1/6 = 3/6.
  4. 4Simplify 3/6 to its lowest terms.

Answer: 1/2

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Common mistake. Pupils often convert to decimals partway through a multi-step problem, which can introduce rounding errors that compound by the final step. Keep all working in fractions until a final simplified answer is reached.

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