CBSE Class 7 Mathematics Comparing Quantities Ratio Assignment Set 03

VII - Mathematics Assignment No. 01 - Comparing Quantities: Ratios and Proportions

Question 1. Find the ratio of
(a) 50 to 200
Answer: Ratio = \(\frac{50}{200} = \frac{1}{4}\) or \(1:4\)
In simple words: Simplify the fraction by dividing both numbers by their highest common factor, which is 50.

Exam Tip: Always write ratios in their simplest form with quantities expressed in the same units.

Question 1.(b) 50 Paisa to Rs 4
Answer: Convert Rs 4 into Paisa:
Rs 4 = \(4 \times 100 = 400\) Paisa
Ratio = \(\frac{50}{400} = \frac{1}{8}\) or \(1:8\)
In simple words: Convert both values to the same unit (paisa) before comparing them.

Exam Tip: Always ensure both quantities are in identical units (convert rupees to paisa or vice versa) before taking their ratio.

Question 1.(c) 25 minutes to 1 Hour
Answer: Convert 1 hour into minutes:
1 Hour = 60 minutes
Ratio = \(\frac{25}{60} = \frac{5}{12}\) or \(5:12\)
In simple words: Change 1 hour to 60 minutes, then find the ratio with 25 minutes.

Exam Tip: Time quantities must be converted to the smaller unit (minutes) to avoid fractions.

Question 1.(d) 250 ml to 2 litres
Answer: Convert 2 litres into millilitres:
2 litres = \(2 \times 1000 = 2000\) ml
Ratio = \(\frac{250}{2000} = \frac{1}{8}\) or \(1:8\)
In simple words: Convert litres to millilitres since 1 litre equals 1000 millilitres.

Exam Tip: Use standard conversion factors like 1 litre = 1000 ml carefully.

Question 2. Express the following ratios in the simplest form
(a) 60 : 90
Answer: \(\frac{60}{90} = \frac{6}{9} = \frac{2}{3}\) or \(2:3\)
In simple words: Divide both numbers by their highest common factor, which is 30.

Exam Tip: Cancel out common zeros and factors until no common factor other than 1 remains.

Question 2.(b) 100 : 500
Answer: \(\frac{100}{500} = \frac{1}{5}\) or \(1:5\)
In simple words: Divide both numbers by 100.

Exam Tip: Quick cancellation of trailing zeros makes simplification faster.

Question 2.(c) 125 : 500
Answer: \(\frac{125}{500} = \frac{1}{4}\) or \(1:4\)
In simple words: Divide both numbers by 125.

Question 2.(d) 121 : 1331
Answer: \(\frac{121}{1331} = \frac{1}{11}\) or \(1:11\)
In simple words: Divide both numbers by 121, knowing that \(1331 = 11^3\).

Exam Tip: Memorizing squares up to 30 and cubes up to 15 helps instantly with large numbers.

Question 3. Express the following in the language of ratio
(a) The length of a rectangle is twice its width
Answer: Let the width be \(x\). Then the length is \(2x\).
Ratio of length to width = \(\frac{2x}{x} = 2:1\)
In simple words: Length is 2 times width, so the ratio is 2 to 1.

Exam Tip: Pay attention to the order specified: length to width means length comes first in the ratio.

Question 3.(b) The number of students who passed in an examination is \(\frac{3}{4}\) of those who appeared
Answer: Let the number of students who appeared be \(x\).
Passed students = \(\frac{3}{4}x\)
Ratio of passed students to appeared students = \(\frac{\frac{3}{4}x}{x} = \frac{3}{4}\) or \(3:4\)
In simple words: Out of 4 students who appeared, 3 passed, giving a ratio of 3:4.

Exam Tip: Fractional parts directly translate to numerators and denominators of ratios.

Question 4. The number of boys and girls in a school are 2100 and 1800 respectively. Express the ratio of the number of boys to that of girls in the simplest form
Answer: Number of boys = 2100
Number of girls = 1800
Ratio of boys to girls = \(\frac{2100}{1800} = \frac{21}{18} = \frac{7}{6}\) or \(7:6\)
In simple words: Divide both numbers by 300 to get the simplest ratio form.

Exam Tip: The word 'respectively' means the first number belongs to boys and the second to girls.

Question 5. The ratio of Copper and Zinc in an alloy is \(4:3\). Find the weight of each metal in an alloy whose weight is 343 grams.
Answer: Ratio of Copper to Zinc = \(4:3\)
Sum of ratio terms = \(4 + 3 = 7\)
Total weight = 343 grams
Weight of Copper = \(\frac{4}{7} \times 343 = 4 \times 49 = 196\) grams
Weight of Zinc = \(\frac{3}{7} \times 343 = 3 \times 49 = 147\) grams
In simple words: Add the ratio parts to find the total parts, then multiply each metal's fractional share by the total weight.

Exam Tip: Always verify that the sum of the individual weights equals the total given weight (\(196 + 147 = 343\)).

Question 6. Compare the ratio
(a) \(3:4\) and \(5:7\)
Answer: Write as fractions: \(\frac{3}{4}\) and \(\frac{5}{7}\)
LCM of denominators 4 and 7 is 28.
\(\frac{3 \times 7}{4 \times 7} = \frac{21}{28}\)
\(\frac{5 \times 4}{7 \times 4} = \frac{20}{28}\)
Since \(\frac{21}{28} > \frac{20}{28}\), therefore \(\frac{3}{4} > \frac{5}{7}\) or \(3:4 > 5:7\)
In simple words: Make denominators equal using LCM to easily compare which fraction is larger.

Exam Tip: Cross-multiplication (\(3 \times 7 = 21\) and \(4 \times 5 = 20\)) is a quick alternative method for comparing two ratios.

Question 6.(b) \(5:4\) and \(1:3\)
Answer: Write as fractions: \(\frac{5}{4}\) and \(\frac{1}{3}\)
LCM of denominators 4 and 3 is 12.
\(\frac{5 \times 3}{4 \times 3} = \frac{15}{12}\)
\(\frac{1 \times 4}{3 \times 4} = \frac{4}{12}\)
Since \(\frac{15}{12} > \frac{4}{12}\), therefore \(\frac{5}{4} > \frac{1}{3}\) or \(5:4 > 1:3\)
In simple words: Convert to like fractions and compare their numerators.

Exam Tip: An improper fraction is always greater than a proper fraction.

Question 7. Divide 250 in the ratio \(3:2\)
Answer: Sum of ratio terms = \(3 + 2 = 5\)
First part = \(\frac{3}{5} \times 250 = 3 \times 50 = 150\)
Second part = \(\frac{2}{5} \times 250 = 2 \times 50 = 100\)
Required parts are 150 and 100.
In simple words: Divide 250 into 5 equal parts, then assign 3 parts to the first quantity and 2 parts to the second.

Exam Tip: Check your answer by adding the resulting parts (\(150 + 100 = 250\)).

Question 8. If \(A:B = 3:4\) and \(B:C = 5:6\), find \(A:C\)
Answer: Given \(\frac{A}{B} = \frac{3}{4}\) and \(\frac{B}{C} = \frac{5}{6}\)
\(A:C = \frac{A}{C} = \frac{A}{B} \times \frac{B}{C} = \frac{3}{4} \times \frac{5}{6} = \frac{15}{24} = \frac{5}{8}\) or \(5:8\)
In simple words: Multiply the two given ratio fractions to cancel out B and get the ratio of A to C.

Question 9. If \(A:B = 2:3\) and \(B:C = 7:8\), find \(A:B:C\)
Answer: \(A:B = 2:3\)
\(B:C = 7:8\)
To make the common term \(B\) equal in both ratios, find the LCM of 3 and 7, which is 21.
Multiply \(A:B\) by 7: \(A:B = (2 \times 7) : (3 \times 7) = 14:21\)
Multiply \(B:C\) by 3: \(B:C = (7 \times 3) : (8 \times 3) = 21:24\)
Combining them gives \(A:B:C = 14:21:24\)
In simple words: Make the middle term (B) equal in both ratios by scaling them properly, then combine.

Exam Tip: The bridge method (making the middle letter's coefficients equal) is the most reliable way to chain ratios.

Question 10. If \(3A = 2B = 5C\), find \(A:B:C\)
Answer: Let \(3A = 2B = 5C = k\)
Then \(A = \frac{k}{3}\), \(B = \frac{k}{2}\), \(C = \frac{k}{5}\)
\(A:B:C = \frac{1}{3} : \frac{1}{2} : \frac{1}{5}\)
LCM of denominators 3, 2, and 5 is 30.
Multiply each term by 30:
\(A:B:C = \left(\frac{1}{3} \times 30\right) : \left(\frac{1}{2} \times 30\right) : \left(\frac{1}{5} \times 30\right) = 10 : 15 : 6\)
In simple words: Take the reciprocal of the coefficients of A, B, and C, then multiply by the LCM to clear fractions.

Exam Tip: Alternatively, cover the coefficient of A and multiply the other two coefficients (\(2 \times 5 = 10\)) to find A's part quickly.