1956 AMC 12 Problem 19

Question

Two candles of the same height are lighted at the same time. The first is consumed in

4

hours and the second in

3

hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted was the first candle twice the height of the second?

\dfrac34

hr.

1\dfrac12

hr.

2

hr.

2\dfrac25

hr.

2\dfrac12

hr.

Scale the common initial height to

1

After

t

hours the remaining fractions are

1-\frac{t}{4}

and

1-\frac{t}{3}

Let each initial height be

1.

After

t

hours the remaining heights are

1-\frac{t}{4}

and

1-\frac{t}{3}.

The required condition is

1-\frac t4=2\left(1-\frac t3\right).

Multiplying by

12

gives

12-3t=24-8t,

so

5t=12

and

t=\frac{12}{5}=2\dfrac25.

Thus, the correct answer is D.

Problem 19 in other years