Ratio and Proportion

Q1: Which of the following statements are true?

i. \(27 : 36 = 4.5 : 6\)

Step 1: Simplify both ratios: \[ \frac{27}{36} = \frac{3}{4}, \quad \frac{4.5}{6} = \frac{45}{60} = \frac{3}{4} \]Answer: True

ii. \(\frac{3}{4}:\frac{15}{16} = \frac{2}{3}:\frac{5}{6}\)

Step 1: Convert both sides to simple fractions: \[ \text{LHS} = \frac{3}{4} \div \frac{15}{16} = \frac{3}{4} \times \frac{16}{15} = \frac{48}{60} = \frac{4}{5} \\ \text{RHS} = \frac{2}{3} \div \frac{5}{6} = \frac{2}{3} \times \frac{6}{5} = \frac{12}{15} = \frac{4}{5} \]Answer: True

iii. ₹14 : ₹21 = 2 pens : 3 pens

Step 1: Simplify both ratios: \[ \frac{14}{21} = \frac{2}{3}, \quad \frac{2\text{ pens}}{3\text{ pens}} = \frac{2}{3} \]Answer: True

iv. 6.5 km : 2.6 km = ₹60 : ₹24

Step 1: Simplify both ratios: \[ \frac{6.5}{2.6} = \frac{65}{26} = \frac{5}{2}, \quad \frac{60}{24} = \frac{5}{2} \]Answer: True

Q2: Check whether the following numbers are in proportion or not:

i. \(8,\ 12,\ 18,\ 24\)

Step 1: Check if \(\frac{8}{12} = \frac{18}{24}\) \[ \frac{8}{12} = \frac{2}{3},\quad \frac{18}{24} = \frac{3}{4} \] Answer: Not in proportion

ii. \(6.4,\ 3.6,\ 4.8,\ 2.7\)

Step 1: Check if \(\frac{6.4}{3.6} = \frac{4.8}{2.7}\) \[ \frac{6.4}{3.6} = \frac{64}{36} = \frac{16}{9},\quad \frac{4.8}{2.7} = \frac{48}{27} = \frac{16}{9} \] Answer: In proportion

iii. \(11\frac{1}{3},\ 9\frac{1}{3},\ 8\frac{1}{2},\ 7\)

Step 1: Convert to improper fractions: \[ 11\frac{1}{3} = \frac{34}{3},\quad 9\frac{1}{3} = \frac{28}{3},\quad 8\frac{1}{2} = \frac{17}{2},\quad 7 = \frac{14}{2} \] Step 2: Check if \(\frac{34}{28} = \frac{17}{14}\) \[ \frac{34}{28} = \frac{17}{14},\quad \frac{17}{14} = \frac{17}{14} \] Answer: In proportion

iv. \(0.36,\ 1.8,\ 6.4,\ 32\)

Step 1: Check if \(\frac{0.36}{1.8} = \frac{6.4}{32}\) \[ \frac{0.36}{1.8} = \frac{36}{180} = \frac{1}{5},\quad \frac{6.4}{32} = \frac{64}{320} = \frac{1}{5} \] Answer: In proportion

v. \(\frac{3}{4},\ \frac{5}{6},\ \frac{7}{8},\ \frac{9}{10}\)

Step 1: Check if \(\frac{\frac{3}{4}}{\frac{5}{6}} = \frac{\frac{7}{8}}{\frac{9}{10}}\) \[ \frac{3}{4} \div \frac{5}{6} = \frac{3}{4} \times \frac{6}{5} = \frac{18}{20} = \frac{9}{10} \\ \frac{7}{8} \div \frac{9}{10} = \frac{7}{8} \times \frac{10}{9} = \frac{70}{72} = \frac{35}{36} \\ \frac{9}{10} \ne \frac{35}{36} \] Answer: Not in proportion

Q3: Find the value of \(x\) in each of the following:

i. \(8:x \::: 6:27\)

Step 1: Use the property of proportion: \[ \frac{8}{x} = \frac{6}{27} \\ \Rightarrow 8 \times 27 = 6 \times x \\ \Rightarrow 216 = 6x \\ \Rightarrow x = \frac{216}{6} = 36 \]Answer: x = 36

ii. \(5.6:3.5 \::: x:1.25\)

Step 1: Use proportion: \[ \frac{5.6}{3.5} = \frac{x}{1.25} \\ \Rightarrow \frac{56}{35} = \frac{x}{1.25} = \frac{8}{5} \\ \Rightarrow x = \frac{8}{5} \times 1.25 = \frac{8 \times 125}{5 \times 100} = \frac{1000}{500} = 2 \]Answer: x = 2

iii. \(1\frac{4}{5}:2\frac{4}{5} \::: x:3\frac{1}{2}\)

Step 1: Convert to improper fractions: \[ 1\frac{4}{5} = \frac{9}{5}, \quad 2\frac{4}{5} = \frac{14}{5}, \quad 3\frac{1}{2} = \frac{7}{2} \]Step 2: Apply proportion: \[ \frac{9}{5} : \frac{14}{5} = x : \frac{7}{2} \\ \Rightarrow \frac{9}{14} = \frac{x}{\frac{7}{2}} \\ \Rightarrow x = \frac{9}{14} \times \frac{7}{2} = \frac{63}{28} = \frac{9}{4} = 2\frac{1}{4} \]Answer: x = \(2\frac{1}{4}\)

iv. \(\frac{2}{3}:\frac{4}{7} \::: 1\frac{5}{6}:x\)

Step 1: Convert mixed number to improper fraction: \[ 1\frac{5}{6} = \frac{11}{6} \]Step 2: Use proportion: \[ \frac{\frac{2}{3}}{\frac{4}{7}} = \frac{\frac{11}{6}}{x} \\ \Rightarrow \frac{2}{3} \div \frac{4}{7} = \frac{2}{3} \times \frac{7}{4} = \frac{14}{12} = \frac{7}{6} \\ \Rightarrow \frac{11}{6} \div x = \frac{7}{6} \\ \Rightarrow x = \frac{11}{6} \div \frac{7}{6} = \frac{11}{6} \times \frac{6}{7} = \frac{66}{42} = \frac{11}{7} \\ \Rightarrow x = 1\frac{4}{7} \]Answer: x = \(1\frac{4}{7}\)

Q4: Find the fourth proportional to:

i. \(2.8,\ 14 \ and \ 3.5\)

Step 1: Let the fourth proportional be \(x\). \[ \frac{2.8}{14} = \frac{3.5}{x} \\ \Rightarrow 2.8x = 14 \times 3.5 = 49 \\ \Rightarrow x = \frac{49}{2.8} = 17.5 \] Answer: x = 17.5

ii. \(3\frac{1}{3},\ 1\frac{2}{3} \ and\ 2\frac{1}{2}\)

Step 1: Convert to improper fractions: \[ 3\frac{1}{3} = \frac{10}{3},\quad 1\frac{2}{3} = \frac{5}{3},\quad 2\frac{1}{2} = \frac{5}{2} \] Let fourth proportional be \(x\): \[ \frac{10}{3} : \frac{5}{3} = \frac{5}{2} : x \\ \Rightarrow \frac{10}{3} \div \frac{5}{3} = \frac{5}{2} \div x \\ \Rightarrow 2 = \frac{5}{2} \div x \\ \Rightarrow x = \frac{5}{2} \div 2 = \frac{5}{4} = 1\frac{1}{4} \] Answer: x = \(1\frac{1}{4}\)

iii. \(1\frac{5}{7},\ 2\frac{3}{14} \ and \ 3\frac{3}{5}\)

Step 1: Convert to improper fractions: \[ 1\frac{5}{7} = \frac{12}{7},\quad 2\frac{3}{14} = \frac{31}{14},\quad 3\frac{3}{5} = \frac{18}{5} \] Let fourth proportional be \(x\): \[ \frac{12}{7} : \frac{31}{14} = \frac{18}{5} : x \\ \Rightarrow \frac{12}{7} \div \frac{31}{14} = \frac{18}{5} \div x \\ \Rightarrow \frac{12}{7} \times \frac{14}{31} = \frac{18}{5} \div x \\ \Rightarrow \frac{168}{217} = \frac{18}{5} \div x \\ \Rightarrow x = \frac{18}{5} \div \frac{168}{217} = \frac{18}{5} \times \frac{217}{168} = \frac{3906}{840} \] Simplify: \[ \frac{3906}{42} = 93,\quad \frac{840}{42} = 20 \\ \Rightarrow x = \frac{93}{20} = 4\frac{13}{20} \]Answer: x = \(4\frac{13}{20}\)

iv. \(1\frac{1}{5},\ 1\frac{3}{5} \ and \ 2.1\)

Step 1: Convert to decimals: \[ 1\frac{1}{5} = 1.2,\quad 1\frac{3}{5} = 1.6,\quad 2.1 = 2.1 \] Let fourth proportional be \(x\): \[ \frac{1.2}{1.6} = \frac{2.1}{x} \\ \Rightarrow 1.2x = 1.6 \times 2.1 = 3.36 \\ \Rightarrow x = \frac{3.36}{1.2} = 2.8 \]Answer: x = 2.8

Q5: Find the third proportional to:

i. 12, 16

Step 1: Let third proportional = \(x\)
By definition: \[ \frac{12}{16} = \frac{16}{x} \\ \Rightarrow 12x = 256 \\ \Rightarrow x = \frac{256}{12} = 21\frac{1}{3} \]Answer: \(x = 21\frac{1}{3}\)

ii. 4.5, 6

Step 1: Let third proportional = \(x\) \[ \frac{4.5}{6} = \frac{6}{x} \\ \Rightarrow 4.5x = 36 \\ \Rightarrow x = \frac{36}{4.5} = 8 \]Answer: \(x = 8\)

iii. \(5\frac{1}{2}, 16\frac{1}{2}\)

Step 1: Convert to improper fractions: \[ 5\frac{1}{2} = \frac{11}{2},\quad 16\frac{1}{2} = \frac{33}{2} \] Let third proportional = \(x\): \[ \frac{11}{2} : \frac{33}{2} = \frac{33}{2} : x \\ \Rightarrow \frac{11}{2} \div \frac{33}{2} = \frac{33}{2} \div x \\ \Rightarrow \frac{1}{3} = \frac{33}{2} \div x \\ \Rightarrow x = \frac{33}{2} \div \frac{1}{3} = \frac{33}{2} \times 3 = \frac{99}{2} = 49\frac{1}{2} \]Answer: \(x = 49\frac{1}{2}\)

iv. \(3\frac{1}{2}, 8\frac{3}{4}\)

Step 1: Convert to improper fractions: \[ 3\frac{1}{2} = \frac{7}{2},\quad 8\frac{3}{4} = \frac{35}{4} \] Let third proportional = \(x\) \[ \frac{7}{2} : \frac{35}{4} = \frac{35}{4} : x \\ \\ \Rightarrow \frac{7}{2} \div \frac{35}{4} = \frac{35}{4} \div x \\ \\ \Rightarrow \frac{7}{2} \times \frac{4}{35} = \frac{35}{4} \times \frac{1}{x}\\ \\ \Rightarrow \frac{14}{35} = \frac{35}{4x} \\ \\ \Rightarrow \frac{2}{5} = \frac{35}{4x}\\ \\ \Rightarrow x = \frac{35 \times 5}{2 \times 4} = \frac{175}{8} = 21\frac{7}{8} \]Answer: \(x = 21\frac{7}{8}\)

Q6: Find the mean proportion between:

i. \(8\ and\ 18\)

Step 1: Mean proportion = \(\sqrt{8 \times 18} = \sqrt{144} = 12\)
Answer: 12

ii. \(0.3\ and\ 2.7\)

Step 1: Mean proportion = \(\sqrt{0.3 \times 2.7} = \sqrt{0.81} = 0.9\)
Answer: 0.9

iii. \(66\frac{2}{3}\ and\ 6\)

Step 1: Convert to improper fraction: \(66\frac{2}{3} = \frac{200}{3}\)
Mean proportion = \(\sqrt{\frac{200}{3} \times 6} = \sqrt{400} = 20\)
Answer: 20

iv. \(1.25\ and\ 0.45\)

Step 1: Mean proportion = \(\sqrt{1.25 \times 0.45} = \sqrt{0.5625} = 0.75\)
Answer: 0.75

v. \(\frac{1}{7}\ and\ \frac{4}{63}\)

Step 1: Mean proportion = \[ \sqrt{\frac{1}{7} \times \frac{4}{63}} = \sqrt{\frac{4}{441}} = \frac{2}{21} \]Answer: \(\frac{2}{21}\)

Q7: If 28 is the third proportional to 7 and \(x\), find the value of \(x\).

Third proportional means:
If \(a : b = b : c\), then \(c\) is called the third proportional to \(a\) and \(b\).
Here, \(a = 7,\ b = x,\ c = 28\)
Step 1: Apply the third proportional rule: \[ \frac{7}{x} = \frac{x}{28} \]Step 2: Cross-multiply: \[ 7 \times 28 = x \times x \\ \Rightarrow 196 = x^2 \]Step 3: Take square root of both sides: \[ x = \sqrt{196} = 14 \]Answer: x = 14

Q8: If 18, x, 50 are in continued proportion, find the value of x.

If three numbers \(a,\ b,\ c\) are in continued proportion, then: \[ \frac{a}{b} = \frac{b}{c} \\ \Rightarrow b^2 = a \cdot c \]Step 1: Here, \[ a = 18,\quad b = x,\quad c = 50 \\ \Rightarrow x^2 = 18 \times 50 = 900 \]Step 2: Take square root on both sides: \[ x = \sqrt{900} = 30 \]Answer: x = 30

Q9: A rod was cut into two pieces in the ratio 7 : 5. If the length of the smaller piece was 45.5 cm, then find the length of the longer piece.

Ratio of the two pieces = 7 : 5
Smaller part = 5 parts = 45.5 cm
Step 1: Find the value of 1 part: \[ \text{1 part} = \frac{45.5}{5} = 9.1\ \text{cm} \]Step 2: Now find the longer piece (7 parts): \[ \text{Longer piece} = 9.1 \times 7 = 63.7\ \text{cm} \]Answer: The length of the longer piece is 63.7 cm.

Q10: The area of two rectangular fields are in the ratio 5 : 9. Find the area of the smaller field if that of the larger field is 2331 sq. metres.

Ratio of areas = 5 : 9
Area of the larger field = 2331 sq. metres
Step 1: Let 9 parts = 2331 sq. metres
So, 1 part = \(\frac{2331}{9} = 259\) sq. metres
Step 2: Area of the smaller field = 5 parts \[ = 259 \times 5 = 1295\ \text{sq. metres} \]Answer: Area of the smaller field is 1295 sq. metres.

Q11: What number must be subtracted from each of the numbers 41, 55, 36, 48, so that the differences are proportional?

Let the number to be subtracted be \(x\).
Then the four numbers become: \[ 41 – x,\quad 55 – x,\quad 36 – x,\quad 48 – x \]Step 1: According to the question, \[ (41 – x):(55 – x) = (36 – x):(48 – x) \]Step 2: Convert into equation using cross multiplication: \[ (41 – x)(48 – x) = (55 – x)(36 – x) \]Step 3: Expand both sides:
Left Side: \[ (41 – x)(48 – x) = 1968 – 89x + x^2 \] Right Side: \[ (55 – x)(36 – x) = 1980 – 91x + x^2 \]Step 4: Equate both sides: \[ 1968 – 89x + x^2 = 1980 – 91x + x^2 \]Subtract \(x^2\) from both sides: \[ 1968 – 89x = 1980 – 91x \]Step 5: Simplify:
Bring like terms together: \[ -89x + 91x = 1980 – 1968 \\ \Rightarrow 2x = 12 \\ \Rightarrow x = 6 \]Answer: 6 must be subtracted from each number.

Q12: An alloy is to contain copper and zinc in the ratio 9 : 4. Find the quantity of zinc to be melted with \(2\frac{2}{5}\) kg of copper, to get the desired alloy.

Copper : Zinc = 9 : 4
Copper = \(2\frac{2}{5}\) kg = \(\frac{12}{5}\) kg
Let quantity of zinc be \(x\) kg
Step 1: Use the ratio: \[ \frac{\text{Copper}}{\text{Zinc}} = \frac{9}{4} \\ \Rightarrow \frac{12}{5x} = \frac{9}{4} \]Step 2: Cross multiply: \[ 12 \times 4 = 9 \times 5x \\ \Rightarrow 48 = 45x \\ \Rightarrow x = \frac{48}{45} = \frac{16}{15} \]Step 3: Convert to mixed number: \[ \frac{16}{15} = 1\frac{1}{15}\ \text{kg} \]Answer: Zinc to be added = \(1\frac{1}{15}\) kg