Fête Bottle Sizing and a Decimal Claim
At a school fête, three drink dispensers hold 7,560 mL, 5,040 mL, and 6,300 mL of lemonade. Volunteers want to pour all of it into identical capped bottles, using one dispenser at a time so flavours do not mix. Every bottle must be completely full, and no lemonade may be left over. A volunteer suggests 1.5 L bottles because 'they are standard and will divide all three amounts exactly.' Another volunteer wants the largest possible bottle size to reduce bottle count. Later, a student writes the ratio of the first dispenser to the second as 7560:5040 and claims that because both numbers are integers, the ratio cannot be written as a terminating decimal. The fête committee needs a bottle decision before ordering labels, and the student's claim is being used to argue against printing decimal capacities on the labels.
1. Find the largest bottle capacity, in millilitres, that fills every bottle exactly with no lemonade left over, and state how many bottles are needed in total.
Read the worked answer
The capacity must divide 7,560, 5,040 and 6,300 exactly, so it is the HCF of the three amounts.
Prime factors:
- 7,560 = 2³ × 3³ × 5 × 7
- 5,040 = 2⁴ × 3² × 5 × 7
- 6,300 = 2² × 3² × 5² × 7
Take the lowest power of each common prime: 2² × 3² × 5 × 7 = 4 × 9 × 35 = 1,260 mL.
Bottle counts: 7,560 ÷ 1,260 = 6; 5,040 ÷ 1,260 = 4; 6,300 ÷ 1,260 = 5. Total = 6 + 4 + 5 = 15 bottles.
2. Test the suggestion that 1.5 L bottles divide all three amounts exactly. Show the division and explain whether the suggestion is usable.
Read the worked answer
1.5 L = 1,500 mL. Check each dispenser:
- 7,560 ÷ 1,500 = 5.04
- 5,040 ÷ 1,500 = 3.36
- 6,300 ÷ 1,500 = 4.2
None is a whole number, so full 1.5 L bottles would leave lemonade over. The suggestion is not usable under the no-waste rule. A useful check is that 1,500 must be a common divisor of all three amounts; it is not even a divisor of 7,560.
3. Evaluate the student's claim that 7560:5040 cannot be written as a terminating decimal because both numbers are integers. Give the simplified ratio and the decimal form.
Read the worked answer
The claim confuses 'integer' with 'non-terminating'. The ratio is a rational number, and many rational numbers have terminating decimals.
Simplify 7,560:5,040 by dividing both terms by 2,520: 7,560 ÷ 2,520 = 3; 5,040 ÷ 2,520 = 2, so the ratio is 3:2.
As a quotient: 7,560/5,040 = 3/2 = 1.5, which terminates after one decimal place. In general, a fraction in lowest terms terminates exactly when its denominator has no prime factors other than 2 and 5; here the denominator is 2.