Question
If x : y = 3 : 4, find (3x + 4y) : (5x + 6y).
✓
Answer
$\text{x : y}=3:4$
$\Rightarrow\frac{\text{x}}{\text{y}}=\frac{3}{4}$
Now $(3\text{x}+4\text{y}):(5\text{x}+6\text{y})$
$\Rightarrow\frac{3\text{x}+4\text{y}}{5\text{x}+6\text{y}}=\frac{3\frac{\text{x}}{\text{y}}+4\frac{\text{y}}{\text{y}}}{5\frac{\text{x}}{\text{y}}+6\frac{\text{y}}{\text{y}}}$
(Dividing each term by y)
$=\frac{3\frac{\text{x}}{\text{y}}+4}{5\frac{\text{x}}{\text{y}}+6}=\frac{3\times\frac{3}{4}+4}{5\times\frac{3}{4}+6}$
$\Big(\because\frac{\text{x}}{\text{y}}=\frac{3}{4}\Big)$
$=\frac{\frac{9}{4}+4}{\frac{15}{4}+6}=\frac{\frac{9+16}{4}}{\frac{15+24}{4}}$
$=\frac{25}{4}\times\frac{4}{39}=\frac{25}{39}$
$\therefore(3\text{x}+4\text{y}):(5\text{x}+6\text{y})=25:39$
$\Rightarrow\frac{\text{x}}{\text{y}}=\frac{3}{4}$
Now $(3\text{x}+4\text{y}):(5\text{x}+6\text{y})$
$\Rightarrow\frac{3\text{x}+4\text{y}}{5\text{x}+6\text{y}}=\frac{3\frac{\text{x}}{\text{y}}+4\frac{\text{y}}{\text{y}}}{5\frac{\text{x}}{\text{y}}+6\frac{\text{y}}{\text{y}}}$
(Dividing each term by y)
$=\frac{3\frac{\text{x}}{\text{y}}+4}{5\frac{\text{x}}{\text{y}}+6}=\frac{3\times\frac{3}{4}+4}{5\times\frac{3}{4}+6}$
$\Big(\because\frac{\text{x}}{\text{y}}=\frac{3}{4}\Big)$
$=\frac{\frac{9}{4}+4}{\frac{15}{4}+6}=\frac{\frac{9+16}{4}}{\frac{15+24}{4}}$
$=\frac{25}{4}\times\frac{4}{39}=\frac{25}{39}$
$\therefore(3\text{x}+4\text{y}):(5\text{x}+6\text{y})=25:39$
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