Two positive numbers x and y are in the ratio a:b, where 0< a< b. If x+y=c, then the smaller of x and y is

1982 AMC 12 Problem 5

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Two positive numbers

x

and

y

are in the ratio

a:b,

where

0\lt a\lt b.

x+y=c,

then the smaller of

x

and

y

\frac{ac}{b}

\frac{bc-ac}{b}

\frac{ac}{a+b}

\frac{bc}{a+b}

\frac{ac}{b-a}

Write the numbers as

ka

and

kb

Use their sum to determine the common scale factor

Write

x=ka

and

y=kb.

Since

a\lt b,

x

is smaller. From

k(a+b)=c,

k=\frac{c}{a+b},

x=\frac{ac}{a+b}.

Therefore, the correct answer is C.