Optional enrichment: Remaining fractions, rates, recurring decimals and rational relationships
Example 1
Pump A takes x hours to fill a tank; pump B takes 5 hours longer. Starting with an empty tank, A runs alone for 4 hours. B then joins A, and they finish filling the tank in a further 2370 hours. Find x, with x>4. Both pumps run at constant rates.
Show worked answer
Answer: 9
- The filled fractions give x4+(2370)(x1+x+51)=1.
- Clear the denominators and solve the quadratic;
- its positive root is x=9 hours.
Example 2
Two positive numbers differ by 6. The reciprocal of the smaller minus the reciprocal of the larger is 203. Find the larger number.
Show worked answer
Answer: 10
- Let the larger be x.
- Then x−61−x1=203.
- Combine the fractions and solve the resulting quadratic.
- The root with x>6 is 10.
Example 3
Pump A takes x hours to fill a tank; pump B takes 2 hours longer. Starting with an empty tank, A runs alone for 3 hours. B then joins A, and they finish filling the tank in a further 67 hours. Find x, with x>3. Both pumps run at constant rates.
Show worked answer
Answer: 5
- The filled fractions give x3+(67)(x1+x+21)=1.
- Clear the denominators and solve the quadratic;
- its positive root is x=5 hours.