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Home » Mathematics » Middle School Classroom Math (Ages 9-13)
 
A decreasing ratio
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#1
A decreasing ratio
John was three times as old as his sister 2 years ago and five times as old 2 years before that. In how many years will the ratio be 2 to 1 ?
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#2
Solution
Let j be John's current age and s be his sister's current age. We have:

\frac {j - 2}{s - 2} = 3 \\ \\ \frac {j - 4}{s - 4} = 5

Solving the first, we have:

j - 2 = 3s - 6 \\ \\ j = 3s - 4

And the second:

j - 4 = 5s - 20 \\ \\ j = 5s - 16

Substituting:

3s - 4 = 5s - 16 \\ \\ - 2s = - 12 \\ \\ s = 6

Plugging back in:

j = 3s - 4 \\ \\ j = 3 \cdot 6 - 4 \\ \\ j = 14

So John is 14 years old and his sister is 6 right now. To see when the ratio becomes double, we will let x be the number of years from now:

\frac {14 + x}{6 + x} = 2 \\ \\ 14 + x = 12 + 2x \\ \\ x = 2

So the ratio is 2 to 1 in \boxed{2} years.
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