We will learn how to divide a number into two parts in a given ratio (i.e., dividing into a given ratio).
Let the number be M. It is to be divided into two parts in the ratio a : b.
The two parts are x and y if x + y = M ........................... (i)
and \(\frac{x}{y}\) = \(\frac{a}{b}\) ........................... (ii)
From (ii), \(\frac{x}{a}\) = \(\frac{y}{b}\) = k (say).
Then , x = ak, y = bk
Substituting in (i), ak + bk = M
⟹ (a + b)k = M
⟹ k = \(\frac{M}{a + b}\)
Therefore, x = ak = \(\frac{a}{a + b}\) M and y = bk = \(\frac{b}{a + b}\) M
Two parts of M in the ratio a : b are \(\frac{aM}{a + b}\) and \(\frac{bM}{a + b}\)
Solved examples on dividing a number into a given ratio:
1. Divide 60 into two parts in the ratio 2 : 3.
Solution:
The two parts are \(\frac{2}{2 + 3}\) × 60 and \(\frac{3}{2 + 3}\) × 60
i.e., \(\frac{2}{5}\) × 60 and \(\frac{3}{5}\) × 60
i.e., 24 and 36
2. Divide 75 into two parts in the ratio 8 : 7
Solution:
The two parts are \(\frac{8}{8 + 7}\) × 75 and \(\frac{7}{8 + 7}\) × 75
i.e., \(\frac{8}{15}\) × 75 and \(\frac{7}{15}\) × 75
i.e., 40 and 35
`● Ratio and proportion
10th Grade Math
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