Question

Mr. Peter gave his eldest son David a bag with 1000 gold coins. David took 230 gold coins from the bag and gave the rest to his younger brothers John, Joe and Jonathan, and advised them to distribute the balance left in the bag amongst themselves in proportion to their age which together amounted to 17.5 years. After a lot of deliberation and discussion John, Joe and Jonathan came to a conclusion to distribute the gold coins. Their methodology was as follows: As often John took 4 gold coins, Joe took 3. As often John took 6 gold coins Jonathan took 7. What was the age of John, Joe and Jonathan?

• A. 6 years, 4.5 years and 7 years respectively
• B. 5 years, 5.5 years and 7 years respectively
• C. 6 years, 5 years and 6.5 years respectively
• D. 5 years, 6.5 years and 6 years respectively

Hint:

Combine the two coin sharing rates into one ratio by matching John's number in both, then apply that same ratio to the ages.

Answer 1

The Correct Option is A

Step 1: Write the two exchange rates as algebra instead of scaling by hand.
Let John's coin share be written two ways using the two given rates. From John:Joe = 4:3, write John's share as $4a$ and Joe's share as $3a$. From John:Jonathan = 6:7, write John's share as $6b$ and Jonathan's share as $7b$.

Step 2: Link the two expressions for John's share.
Since both expressions describe the same John, $4a = 6b$, which simplifies to $\frac{a}{b} = \frac{3}{2}$. So we can write $a = 3k$ and $b = 2k$ for some common number $k$.

Step 3: Get everyone's share in terms of $k$.
John = $4a = 12k$, Joe = $3a = 9k$, Jonathan = $7b = 14k$. This gives the same combined ratio, 12 : 9 : 14, but reached through substitution rather than direct scaling.

Step 4: Use the age total to solve for $k$.
Since ages follow this same ratio and add up to 17.5 years, $12k + 9k + 14k = 17.5$, so $35k = 17.5$, giving $k = 0.5$.

Final Answer:
John's age = $12(0.5) = 6$, Joe's age = $9(0.5) = 4.5$, Jonathan's age = $14(0.5) = 7$ years. \[ \boxed{6,\ 4.5,\ 7 \text{ years}} \]