Question
If $x : y = 6 : 11$, find $(8x - 3y) : (3x + 2y).$
✓
Answer
$\text{x : y}=6:11$
$\frac{\text{x}}{\text{y}}=\frac{6}{11}$ Now $(8\text{x} - 3\text{y}) : (3\text{x} + 2\text{y})$
$=\frac{8\text{x}-3\text{y}}{3\text{x}+2\text{y}}=\frac{8\frac{\text{x}}{\text{y}}-3\frac{\text{y}}{\text{y}}}{3\frac{\text{x}}{\text{y}}+2\frac{\text{y}}{\text{y}}}$ (Dividing each term by $y)$
$=\frac{8\frac{\text{x}}{\text{y}}-3}{3\frac{\text{x}}{\text{y}}+2}=\frac{8\times\frac{6}{11}-3}{3\times\frac{6}{11}+2}$
$\Big(\because\frac{\text{x}}{\text{y}}=\frac{6}{11}\Big)$
$=\frac{\frac{48}{11}+3}{\frac{18}{11}+2}=\frac{\frac{48-33}{11}}{\frac{18+22}{11}}$
$=\frac{\frac{15}{11}}{\frac{40}{11}}=\frac{15}{11}\times\frac{11}{40}=\frac{3}{8}$
$(8\text{x}-3\text{y}):(3\text{x}+2\text{y})=3:8$
$\frac{\text{x}}{\text{y}}=\frac{6}{11}$ Now $(8\text{x} - 3\text{y}) : (3\text{x} + 2\text{y})$
$=\frac{8\text{x}-3\text{y}}{3\text{x}+2\text{y}}=\frac{8\frac{\text{x}}{\text{y}}-3\frac{\text{y}}{\text{y}}}{3\frac{\text{x}}{\text{y}}+2\frac{\text{y}}{\text{y}}}$ (Dividing each term by $y)$
$=\frac{8\frac{\text{x}}{\text{y}}-3}{3\frac{\text{x}}{\text{y}}+2}=\frac{8\times\frac{6}{11}-3}{3\times\frac{6}{11}+2}$
$\Big(\because\frac{\text{x}}{\text{y}}=\frac{6}{11}\Big)$
$=\frac{\frac{48}{11}+3}{\frac{18}{11}+2}=\frac{\frac{48-33}{11}}{\frac{18+22}{11}}$
$=\frac{\frac{15}{11}}{\frac{40}{11}}=\frac{15}{11}\times\frac{11}{40}=\frac{3}{8}$
$(8\text{x}-3\text{y}):(3\text{x}+2\text{y})=3:8$
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