Exercise : Ratios & Proportions - General Questions
✔ Ratios & Proportions -
General Questions
1.
Simplify the ratio 24 : 36 to its simplest form.
View Answer
Answer: Option A
Explanation:
Explanation:
To simplify a ratio, we divide both terms by their Greatest Common Factor (GCF).
Step 1: Find the factors of 24 and 36. The GCF is 12.
Step 2: Divide both terms by 12.
\(24 \div 12 = 2\)
\(36 \div 12 = 3\)
Step 3: The simplified ratio is 2 : 3.
Step 1: Find the factors of 24 and 36. The GCF is 12.
Step 2: Divide both terms by 12.
\(24 \div 12 = 2\)
\(36 \div 12 = 3\)
Step 3: The simplified ratio is 2 : 3.
2.
Find the fourth proportional to the numbers 4, 9, and 12.
View Answer
Answer: Option B
Explanation:
Explanation:
In a proportion, the product of the means equals the product of the extremes.
Step 1: Let the fourth proportional be \(x\). The proportion is written as \(4 : 9 = 12 : x\).
Step 2: Express as fractions: \(\frac{4}{9} = \frac{12}{x}\).
Step 3: Cross-multiply: \(4x = 9 \times 12\).
Step 4: Solve for \(x\): \(4x = 108 \implies x = \frac{108}{4} = 27\).
Step 1: Let the fourth proportional be \(x\). The proportion is written as \(4 : 9 = 12 : x\).
Step 2: Express as fractions: \(\frac{4}{9} = \frac{12}{x}\).
Step 3: Cross-multiply: \(4x = 9 \times 12\).
Step 4: Solve for \(x\): \(4x = 108 \implies x = \frac{108}{4} = 27\).
3.
Divide $1,200 between A and B in the ratio 5 : 7. What are their respective shares?
View Answer
Answer: Option A
Explanation:
Explanation:
Step 1: Calculate the total number of parts in the ratio: \(5 + 7 = 12\).
Step 2: Determine the value of one part: \(\frac{1200}{12} = 100\).
Step 3: Calculate A's share: \(5 \times 100 = 500\).
Step 4: Calculate B's share: \(7 \times 100 = 700\).
Step 2: Determine the value of one part: \(\frac{1200}{12} = 100\).
Step 3: Calculate A's share: \(5 \times 100 = 500\).
Step 4: Calculate B's share: \(7 \times 100 = 700\).
4.
If \(A : B = 2 : 3\) and \(B : C = 4 : 5\), find the combined ratio \(A : B : C\).
View Answer
Answer: Option B
Explanation:
Explanation:
To combine ratios, the common term (B) must have the same value in both ratios.
Step 1: Multiply the first ratio by 4: \(A : B = 8 : 12\).
Step 2: Multiply the second ratio by 3: \(B : C = 12 : 15\).
Step 3: Now that B is 12 in both, combine them: \(A : B : C = 8 : 12 : 15\).
Step 1: Multiply the first ratio by 4: \(A : B = 8 : 12\).
Step 2: Multiply the second ratio by 3: \(B : C = 12 : 15\).
Step 3: Now that B is 12 in both, combine them: \(A : B : C = 8 : 12 : 15\).
5.
What number must be added to each term of the ratio 7 : 13 so that it becomes 2 : 3?
View Answer
Answer: Option B
Explanation:
Explanation:
Step 1: Set up the equation where \(x\) is the number added: \(\frac{7 + x}{13 + x} = \frac{2}{3}\).
Step 2: Cross-multiply: \(3(7 + x) = 2(13 + x)\).
Step 3: Expand: \(21 + 3x = 26 + 2x\).
Step 4: Solve for \(x\): \(3x - 2x = 26 - 21 \implies x = 5\).
Step 2: Cross-multiply: \(3(7 + x) = 2(13 + x)\).
Step 3: Expand: \(21 + 3x = 26 + 2x\).
Step 4: Solve for \(x\): \(3x - 2x = 26 - 21 \implies x = 5\).
6.
Find the mean proportional between 9 and 16.
View Answer
Answer: Option A
Explanation:
Explanation:
The mean proportional between two numbers \(a\) and \(b\) is given by \(\sqrt{ab}\).
Step 1: Multiply the two numbers: \(9 \times 16 = 144\).
Step 2: Take the square root: \(\sqrt{144} = 12\).
Step 3: The mean proportional is 12.
Step 1: Multiply the two numbers: \(9 \times 16 = 144\).
Step 2: Take the square root: \(\sqrt{144} = 12\).
Step 3: The mean proportional is 12.
7.
The ratio of boys to girls in a class is 4 : 5. If there are 20 boys, find the number of girls.
View Answer
Answer: Option A
Explanation:
Explanation:
Step 1: Let the number of boys be \(4x\) and girls be \(5x\).
Step 2: Given \(4x = 20\), solve for \(x\): \(x = 5\).
Step 3: Calculate the number of girls: \(5x = 5 \times 5 = 25\).
Step 2: Given \(4x = 20\), solve for \(x\): \(x = 5\).
Step 3: Calculate the number of girls: \(5x = 5 \times 5 = 25\).
8.
Two numbers are in the ratio 3 : 5. If their sum is 64, find the larger number.
View Answer
Answer: Option B
Explanation:
Explanation:
Step 1: Let the numbers be \(3x\) and \(5x\).
Step 2: Set up the sum equation: \(3x + 5x = 64 \implies 8x = 64\).
Step 3: Solve for \(x\): \(x = 8\).
Step 4: The larger number is \(5x\), so \(5 \times 8 = 40\).
Step 2: Set up the sum equation: \(3x + 5x = 64 \implies 8x = 64\).
Step 3: Solve for \(x\): \(x = 8\).
Step 4: The larger number is \(5x\), so \(5 \times 8 = 40\).
9.
The ratio of the ages of A and B is 3 : 4. After 5 years, the ratio becomes 4 : 5. Find their present ages.
View Answer
Answer: Option A
Explanation:
Explanation:
Step 1: Let present ages be \(3x\) and \(4x\).
Step 2: After 5 years, ages will be \(3x + 5\) and \(4x + 5\).
Step 3: Set up the ratio: \(\frac{3x + 5}{4x + 5} = \frac{4}{5}\).
Step 4: Cross-multiply: \(15x + 25 = 16x + 20 \implies x = 5\).
Step 5: Present ages: \(3(5) = 15\) and \(4(5) = 20\).
Step 2: After 5 years, ages will be \(3x + 5\) and \(4x + 5\).
Step 3: Set up the ratio: \(\frac{3x + 5}{4x + 5} = \frac{4}{5}\).
Step 4: Cross-multiply: \(15x + 25 = 16x + 20 \implies x = 5\).
Step 5: Present ages: \(3(5) = 15\) and \(4(5) = 20\).
10.
A bag contains $1, 50p, and 25p coins in the ratio 5 : 6 : 8. If the total value is $210, find the number of 50p coins.
View Answer
Answer: Option B
Explanation:
Explanation:
Step 1: Determine the value ratio of the coins. $1 = 1.0, 50p = 0.5, 25p = 0.25.
Value ratio = \((5 \times 1) : (6 \times 0.5) : (8 \times 0.25) = 5 : 3 : 2\).
Step 2: Total value parts = \(5 + 3 + 2 = 10\).
Step 3: Value of 50p coins = \(\frac{3}{10} \times 210 = 63\).
Step 4: Number of 50p coins = \(\frac{63}{0.5} = 126\).
Value ratio = \((5 \times 1) : (6 \times 0.5) : (8 \times 0.25) = 5 : 3 : 2\).
Step 2: Total value parts = \(5 + 3 + 2 = 10\).
Step 3: Value of 50p coins = \(\frac{3}{10} \times 210 = 63\).
Step 4: Number of 50p coins = \(\frac{63}{0.5} = 126\).