4 Mark Question

Question 14 Marks

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)
$=\frac{8\frac{\text{x}}{\text{y}}-3}{3\frac{\text{x}}{\text{y}}+2}=\frac{8\times\frac{6}{11}-3}{3\times\frac{6}{11}+2}$
$\Big(\because\frac{\text{x}}{\text{y}}=\frac{6}{11}\Big)$
$=\frac{\frac{48}{11}+3}{\frac{18}{11}+2}=\frac{\frac{48-33}{11}}{\frac{18+22}{11}}$
$=\frac{\frac{15}{11}}{\frac{40}{11}}=\frac{15}{11}\times\frac{11}{40}=\frac{3}{8}$
$(8\text{x}-3\text{y}):(3\text{x}+2\text{y})=3:8$

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Question 24 Marks

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)
$=\frac{3\frac{\text{x}}{\text{y}}+4}{5\frac{\text{x}}{\text{y}}+6}=\frac{3\times\frac{3}{4}+4}{5\times\frac{3}{4}+6}$
$\Big(\because\frac{\text{x}}{\text{y}}=\frac{3}{4}\Big)$
$=\frac{\frac{9}{4}+4}{\frac{15}{4}+6}=\frac{\frac{9+16}{4}}{\frac{15+24}{4}}$
$=\frac{25}{4}\times\frac{4}{39}=\frac{25}{39}$
$\therefore(3\text{x}+4\text{y}):(5\text{x}+6\text{y})=25:39$

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Question 34 Marks

Arrange the following ratios in ascending order:
(5 : 6), (8 : 9), (11 : 18)

Answer

$(5 : 6), (8 : 9), (11 : 18)$
Or $\frac{5}{6},\ \frac{8}{9},\ \frac{11}{18}$
LCM of 6, 9, 18 = 18
$\therefore\frac{5}{6}=\frac{5\times3}{6\times3}=\frac{15}{18}$
$\frac{8}{9}=\frac{8\times2}{9\times2}=\frac{16}{18}$
$\frac{11}{18}=\frac{11}{18}$
We see that:
$\frac{11}{18}<\frac{15}{18}<\frac{16}{18}$
$\therefore(11:18)<(5:6), (8:9)$

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Question 44 Marks

Arrange the following ratios in ascending order:
(11 : 14), (17 : 21), (5 : 7) and (2 : 3)

Answer

$(11:14),(17 :21),(5:7),(2:3)$
Or $\frac{11}{14},\frac{17}{21},\frac{5}{7},\frac{2}{3}$
LCM of 14, 21, 7, 3 = 42
$\begin{array}{c|c} 3 & 14, 21, 7, 3 \\ \hline 7 & 14, 7, 7, 1\\ \hline&2, 1, 1, 1 \end{array}$
LCM = 3 × 7 × 2 = 42
$\therefore\frac{11}{14}=\frac{11\times3}{14\times3}=\frac{33}{42}$
$\frac{17}{21}=\frac{17\times2}{21\times2}=\frac{34}{42}$
$\frac{5}{7}=\frac{5\times6}{7\times6}=\frac{30}{42}$
$\frac{2}{3}=\frac{2\times14}{3\times14}=\frac{28}{42}$
We see that:
$\frac{28}{42}<\frac{30}{42}<\frac{33}{42}<\frac{34}{42}$
$\therefore(2:3)<(5:7)<(11:14)<(17:21)$

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