Henry Ernest Dudeney/Puzzles and Curious Problems/45 - Brother and Sister/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $45$

Brother and Sister

A boy on being asked the age of himself and his sister replied:

"Three years ago I was seven times as old as my sister;

two years ago I was four times as old;

last year I was three times as old;

and this year I am two and one-half times as old."

What are their ages?

Solution

The boy is $10$, and his sister is $4$.

Proof

You need only two of the given statements to solve this.

Let $b$ and $s$ be the ages of the brother and sister respectively.

We have:

\(\ds b - 3\)

\(=\)

\(\ds 7 \paren {s - 3}\)

"Three years ago I was seven times as old as my sister;

\(\ds b - 2\)

\(=\)

\(\ds 4 \paren {s - 2}\)

two years ago I was four times as old;

\(\ds \leadsto \ \ \)

\(\ds b\)

\(=\)

\(\ds 7 s - 18\)

\(\ds \leadsto \ \ \)

\(\ds b\)

\(=\)

\(\ds 4 s - 6\)

\(\ds \leadsto \ \ \)

\(\ds 3 s\)

\(=\)

\(\ds 12\)

\(\ds \leadsto \ \ \)

\(\ds s\)

\(=\)

\(\ds 4\)

\(\ds \leadsto \ \ \)

\(\ds b\)

\(=\)

\(\ds \paren {4 \times 4} - 6\)

\(\ds \)

\(=\)

\(\ds 10\)

The final two statements are redundant.

$\blacksquare$

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $45$. -- Brother and Sister
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $40$. Brother and Sister