28. Age-Proportional Peanut Division
A bag contained 1,000 peanuts. After selling 230 of them, the remaining peanuts were shared among three children—A, B, and C—so that each child received a number of peanuts proportional to his or her age.
The following information about the division is known:
- For every 4 peanuts that A received, B received 6.
- For every 6 peanuts that A received, C received 7.
The sum of their ages is 17.5 years.
Determine:
1. The age of each child.
2. The exact number of peanuts each child received.
Added 9 January 2012
Hint: Translate the two ratio statements into a single three-way ratio for A : B : C, then use the total number of peanuts and total age to scale the ratios.
Solution:
Let the numbers of peanuts received by A, B, and C be in the ratio A : B = 4 : 6 (i.e. 2 : 3) and A : C = 6 : 7. Combining these gives a consistent three-way ratio:
A : B : C = 6 : 9 : 7.
Total peanuts to be shared: 1,000 − 230 = 770.
One “part” therefore equals 770 / (6 + 9 + 7) = 770 / 22 = 35 peanuts.
Peanuts each child received:
A: 6 × 35 = 210
B: 9 × 35 = 315
C: 7 × 35 = 245
Because the peanuts are divided proportional to age, the ages follow the same 6 : 9 : 7 ratio.
Total age = 17.5 years ⇒ one part = 17.5 / 22 = 0.7954545… years.
Ages:
A: 6 × 0.7954545… = 4.7727… years ≈ 4 years 9 months
B: 9 × 0.7954545… = 7.1591… years ≈ 7 years 2 months
C: 7 × 0.7954545… = 5.5682… years ≈ 5 years 7 months
Therefore:
A is about 4 years 9 months old and received 210 peanuts;
B is about 7 years 2 months old and received 315 peanuts;
C is about 5 years 7 months old and received 245 peanuts.
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