CBSE Class 7 Mathematics Comparing Quantities Ratio Assignment Set 01

VII - Mathematics - Assignment No. 03 - Comparing Quantities (Ratio - Proportion)

Question 1. Are 12, 3, 20, 5 in proportion?
Answer: Yes
Product of extremes = \(12 \times 5 = 60\)
Product of means = \(3 \times 20 = 60\)
Since the product of extremes equals the product of means, the given numbers are in proportion.
In simple words: Two ratios are proportional if multiplying the outer numbers gives the same result as multiplying the inner numbers.

Exam Tip: Always verify proportion by checking if \(a : b = c : d\) or if \(a \times d = b \times c\).

Question 2. Find \(x\) if 30, \(x\), 75 and 125 are in proportion.
Answer: 50
Since 30, \(x\), 75, and 125 are in proportion, we have:
\(30 : x = 75 : 125\)
Product of extremes = Product of means
\(30 \times 125 = 75 \times x\)
\(3750 = 75x\)
\(x = \frac{3750}{75} = 50\)
In simple words: Set up the proportion as a fraction equation and cross-multiply to solve for the unknown value \(x\).

Exam Tip: Cross-multiplication is the fastest way to solve for an unknown term in a proportion.

Question 3. Find \(x\) if 6, \(x\), \(x\), 6 are in proportion.
Answer: 6
Since 6, \(x\), \(x\), 6 are in proportion, we have:
\(6 : x = x : 6\)
Product of extremes = Product of means
\(6 \times 6 = x \times x\)
\(36 = x^2\)
\(x = \sqrt{36} = 6\)
In simple words: When four terms are in proportion, the product of the first and fourth equals the product of the second and third.

Exam Tip: If \(a : x = x : d\), then \(x\) is the mean proportional, and \(x = \sqrt{a \times d}\).

Question 4. Find the mean proportional of 4 and 9.
Answer: 6
Let the mean proportional be \(x\). Then 4, \(x\), and 9 are in continued proportion:
\(4 : x = x : 9\)
\(x^2 = 4 \times 9\)
\(x^2 = 36\)
\(x = \sqrt{36} = 6\)
In simple words: The mean proportional between two numbers is the square root of their product.

Exam Tip: Mean proportional between \(a\) and \(b\) is always given by the formula \(\sqrt{ab}\).

Question 5. Find the third proportion of 15, 30.
Answer: 60
Let the third proportion be \(x\). Then 15, 30, and \(x\) are in continued proportion:
\(15 : 30 = 30 : x\)
Product of extremes = Product of means
\(15 \times x = 30 \times 30\)
\(15x = 900\)
\(x = \frac{900}{15} = 60\)
In simple words: For three terms \(a\), \(b\), and \(x\) to be in proportion, the ratio of the first to the second equals the ratio of the second to the third.

Exam Tip: The third proportional to \(a\) and \(b\) is calculated as \(\frac{b^2}{a}\).

Question 6. Find the fourth proportion of 11, 22, 33.
Answer: 66
Let the fourth proportion be \(x\). Then 11, 22, 33, and \(x\) are in proportion:
\(11 : 22 = 33 : x\)
Product of extremes = Product of means
\(11 \times x = 22 \times 33\)
\(11x = 726\)
\(x = \frac{726}{11} = 66\)
In simple words: Arrange the given three numbers in order and use cross-multiplication to find the fourth missing number.

Exam Tip: For numbers \(a\), \(b\), and \(c\), the fourth proportional is \(\frac{b \times c}{a}\).

Question 7. The ratio of length of a ground to its width is 5 : 2. Find the length if the width is 20m.
Answer: 50m
Let the common ratio multiplier be \(x\).
Length = \(5x\)
Width = \(2x\)
Given width = 20m
\(2x = 20\)
\(x = \frac{20}{2} = 10\)
Length = \(5x = 5 \times 10 = 50\text{m}\)
In simple words: Use a variable to represent the parts of the ratio, find the value of one part, and multiply it by the length ratio.

Exam Tip: Whenever quantities are given in a ratio, always introduce a variable like \(x\) to solve word problems easily.

Question 8. Price of 15 articles is Rs. 150. Find the price of 25 articles.
Answer: Rs. 250
Price of 1 article = \(\frac{\text{Rs. } 150}{15} = \text{Rs. } 10\)
Price of 25 articles = \(25 \times \text{Rs. } 10 = \text{Rs. } 250\)
In simple words: First find the unit price by dividing the total cost by the number of items, then multiply by the new quantity.

Exam Tip: Unitary method problems always require finding the value of a single unit first.

Question 9. Cost of 9 tables is Rs. 540. Find the cost of 10 tables.
Answer: Rs. 600
Cost of 1 table = \(\frac{\text{Rs. } 540}{9} = \text{Rs. } 60\)
Cost of 10 tables = \(10 \times \text{Rs. } 60 = \text{Rs. } 600\)
In simple words: Divide the total cost by 9 to get the price of one table, then multiply by 10.

Exam Tip: Double-check division steps before performing multiplication to avoid calculation errors.

Question 10. If 3, 12, and \(x\) are in continued proportion, find \(x\).
Answer: 48
Since 3, 12, and \(x\) are in continued proportion, we have:
\(3 : 12 = 12 : x\)
Product of extremes = Product of means
\(3 \times x = 12 \times 12\)
\(3x = 144\)
\(x = \frac{144}{3} = 48\)
In simple words: In continued proportion with three terms, the middle term is repeated as the second and third ratio terms.

Exam Tip: For three numbers \(a\), \(b\), \(c\) in continued proportion, \(b^2 = a \times c\).