Ratio And Proportion Worksheet - Class 6 Mathematics - FREE PDF Download

Q1: Express each of the following ratios in the simplest form. 
(a) 5.6 m to 28 cm
Ans: 20 : 1
Convert 5.6 m into centimetres: 5.6 × 100 = 560 cm.
So the ratio is 560 cm : 28 cm.
Divide both terms by 28: 560 ÷ 28 = 20 and 28 ÷ 28 = 1.
Therefore the simplified ratio is 20 : 1.

(b) dozen to a score
Ans: 3 : 5
A dozen = 12 and a score = 20.
Ratio = 12 : 20.
Divide both terms by their greatest common divisor 4: 12 ÷ 4 = 3 and 20 ÷ 4 = 5.
So the simplified ratio is 3 : 5.

(c) score to a gross
Ans: 5 : 36
A score = 20 and a gross = 144.
Ratio = 20 : 144.
Divide both terms by 4: 20 ÷ 4 = 5 and 144 ÷ 4 = 36.
Thus the simplified ratio is 5 : 36.

(d) 6 hours to a day
Ans: 1 : 4
A day = 24 hours.
Ratio = 6 : 24.
Divide both terms by 6: 6 ÷ 6 = 1 and 24 ÷ 6 = 4.
So the simplified ratio is 1 : 4.

(e) 20 litres to 0.75 litres
Ans: 80 : 3
Write 0.75 litres as a fraction: 0.75 = 3/4 litre.
Ratio = 20 : 3/4.
Multiply both terms by 4 to remove the fraction: 20 × 4 = 80 and (3/4) × 4 = 3.
Therefore the simplified ratio is 80 : 3.

(f) 1728 : 2400
Ans: 18 : 25
Find a common divisor. Both numbers are divisible by 96.
1728 ÷ 96 = 18 and 2400 ÷ 96 = 25.
Hence the ratio in simplest form is 18 : 25.

Q2: Simplify the following ratios. 
(a) 1/4 : 1/6 : 1/8
Ans: 6 : 4 : 3
Find a common denominator for the fractions. The least common denominator is 24.
Write each fraction with denominator 24: 1/4 = 6/24, 1/6 = 4/24, 1/8 = 3/24.
So the ratio becomes 6 : 4 : 3. This is already in simplest whole-number form.

(b) 3.6 : 4.5
Ans: 4 : 5
Remove decimals by multiplying both numbers by 10: 3.6 × 10 = 36 and 4.5 × 10 = 45.
Ratio = 36 : 45.
Divide both by 9 (their greatest common divisor): 36 ÷ 9 = 4 and 45 ÷ 9 = 5.
So the simplified ratio is 4 : 5.

(c) 3²/₃ : 4¹/₂
Ans: 22 : 27
Convert each mixed number to an improper fraction:
3 2/3 = (3 × 3 + 2)/3 = 11/3.
4 1/2 = (4 × 2 + 1)/2 = 9/2.
Find a common denominator (LCD = 6). Convert both fractions to denominator 6:
11/3 = 22/6 and 9/2 = 27/6.
So the ratio of numerators is 22 : 27.

(d) 3 : 2.4 : 2¹/₄
Ans: 20 : 16 : 15
Convert decimal and mixed number to fractions or remove decimals by a suitable multiplier.
2.4 = 24/10 = 12/5 and 2 1/4 = 9/4.
Choose 20 as a common multiplier to clear denominators: 3 × 20 = 60, 2.4 × 20 = 48, 2.25 × 20 = 45.
So the ratio becomes 60 : 48 : 45.
Divide all terms by their common divisor 3: 60 ÷ 3 = 20, 48 ÷ 3 = 16, 45 ÷ 3 = 15.
Thus the simplified ratio is 20 : 16 : 15.

Q3: Determine if the following ratios form a proportion.
(a) 25 cm : 1 m = $40 :$40:160
(b) 200 ml : 2.5 l = $4 :$4:50
(c) 32 m : 64 m = 7 seconds : 14 seconds
(d) 6.5 litres : 13 litres = 50 kg : 10 kg
Ans: (a, b, c) are proportional; (d) is not proportional.
(a) 25 cm:1 m = 40 : 160
Convert units so they match: 1 m = 100 cm, so 25 cm : 100 cm.
Check by cross-multiplication: 25 × 160 = 4000 and 100 × 40 = 4000.
Since the cross-products are equal, the ratios are in proportion.

(b) 200 ml:2.5 l = 4 : 50
Convert units: 2.5 l = 2500 ml, so 200 ml : 2500 ml.
Check by cross-multiplication: 200 × 50 = 10000 and 2500 × 4 = 10000.
Cross-products are equal, so these ratios are in proportion.

(c) 32 m:64 m = 7 seconds:14 seconds
Units match already. Check by cross-multiplication: 32 × 14 = 448 and 64 × 7 = 448.
Cross-products are equal, so these ratios are in proportion.

(d) 6.5 litres:13 litres = 50 kg:10 kg
Units match for each pair. Check by cross-multiplication: 6.5 × 10 = 65 and 13 × 50 = 650.
Since 65 ≠ 650, the cross-products are not equal. Therefore the ratios are not in proportion.

Q4:  Divide 3000 among P, Q, R in the ratio 2 : 3 : 5
Ans: 600, 900, 1500
Add the parts of the ratio: 2 + 3 + 5 = 10.
Value of one part = 3000 ÷ 10 = 300.
P's share = 2 × 300 = 600.
Q's share = 3 × 300 = 900.
R's share = 5 × 300 = 1500.
So, P = 600, Q = 900, R = 1500.

Q5: Which of the following statements are true?
(a) 25 : 35 = 45 : 55
Ans: False
Check by cross-multiplication: 25 × 55 = 1375 and 35 × 45 = 1575.
Since 1375 ≠ 1575, the statement is false.

(b) 105 : 30 = 49 : 14
Ans: True
Cross-multiply: 105 × 14 = 1470 and 30 × 49 = 1470.
Both products are equal, so the statement is true.

(c) 45 : 48 = 60 : 64
Ans: True
Cross-multiply: 45 × 64 = 2880 and 48 × 60 = 2880.
Since both products are equal, the statement is true.

(d) 2/3 : 7/9 = 3/4 : 5/6
Ans: False
Use cross-products (or multiply across): (2/3) × (5/6) = 10/18 = 5/9, while (7/9) × (3/4) = 21/36 = 7/12.
Since 5/9 ≠ 7/12, the ratios are not equal; the statement is false.

(e) 4.2 : 12.6 = 1.5 : 4.5
Ans: True
Cross-multiply: 4.2 × 4.5 = 18.9 and 12.6 × 1.5 = 18.9.
Since both products are equal, the statement is true.

(f) 12 : 18 = 28 : 12
Ans: False
Cross-multiply: 12 × 12 = 144 and 18 × 28 = 504.
Since 144 ≠ 504, the statement is false.
Therefore the true statements are (b), (c), and (e).