Exercise : Ratios & Proportions - General Questions
✔ Ratios & Proportions -
General Questions
11.
The ratio of incomes of A and B is 5 : 4 and the ratio of their expenditures is 3 : 2. If each saves $1,600, find A's income.
View Answer
Answer: Option B
Explanation:
Explanation:
Step 1: Let incomes be \(5x\) and \(4x\).
Step 2: Savings = Income - Expenditure, so Expenditure = Income - Savings.
Step 3: Set up the expenditure ratio: \(\frac{5x - 1600}{4x - 1600} = \frac{3}{2}\).
Step 4: Cross-multiply: \(10x - 3200 = 12x - 4800\).
Step 5: Solve for \(x\): \(2x = 1600 \implies x = 800\).
Step 6: A's income = \(5 \times 800 = 4000\).
Step 2: Savings = Income - Expenditure, so Expenditure = Income - Savings.
Step 3: Set up the expenditure ratio: \(\frac{5x - 1600}{4x - 1600} = \frac{3}{2}\).
Step 4: Cross-multiply: \(10x - 3200 = 12x - 4800\).
Step 5: Solve for \(x\): \(2x = 1600 \implies x = 800\).
Step 6: A's income = \(5 \times 800 = 4000\).
12.
If \(x : y = 3 : 4\), find the value of \((5x - 2y) : (7x + 2y)\).
View Answer
Answer: Option A
Explanation:
Explanation:
Step 1: Let \(x = 3k\) and \(y = 4k\). For a ratio, we can simplify by using \(x=3, y=4\).
Step 2: Substitute values into the expression: \(\frac{5(3) - 2(4)}{7(3) + 2(4)}\).
Step 3: Calculate: \(\frac{15 - 8}{21 + 8} = \frac{7}{29}\).
Step 4: The ratio is 7 : 29.
Step 2: Substitute values into the expression: \(\frac{5(3) - 2(4)}{7(3) + 2(4)}\).
Step 3: Calculate: \(\frac{15 - 8}{21 + 8} = \frac{7}{29}\).
Step 4: The ratio is 7 : 29.
13.
Three numbers are in the ratio 1 : 2 : 3 and the sum of their squares is 504. Find the numbers.
View Answer
Answer: Option A
Explanation:
Explanation:
Step 1: Let the numbers be \(x, 2x, 3x\).
Step 2: Sum of squares: \(x^2 + (2x)^2 + (3x)^2 = 504\).
Step 3: Simplify: \(x^2 + 4x^2 + 9x^2 = 504 \implies 14x^2 = 504\).
Step 4: Solve for \(x\): \(x^2 = 36 \implies x = 6\).
Step 5: The numbers are \(6, 2(6)=12, 3(6)=18\).
Step 2: Sum of squares: \(x^2 + (2x)^2 + (3x)^2 = 504\).
Step 3: Simplify: \(x^2 + 4x^2 + 9x^2 = 504 \implies 14x^2 = 504\).
Step 4: Solve for \(x\): \(x^2 = 36 \implies x = 6\).
Step 5: The numbers are \(6, 2(6)=12, 3(6)=18\).
14.
The ratio of the volume of two spheres is 8 : 27. Find the ratio of their surface areas.
View Answer
Answer: Option B
Explanation:
Explanation:
Step 1: Volume of a sphere is proportional to \(r^3\). So, \(r_1^3 : r_2^3 = 8 : 27\).
Step 2: Find the radius ratio: \(r_1 : r_2 = \sqrt[3]{8} : \sqrt[3]{27} = 2 : 3\).
Step 3: Surface area is proportional to \(r^2\).
Step 4: Surface area ratio = \(2^2 : 3^2 = 4 : 9\).
Step 2: Find the radius ratio: \(r_1 : r_2 = \sqrt[3]{8} : \sqrt[3]{27} = 2 : 3\).
Step 3: Surface area is proportional to \(r^2\).
Step 4: Surface area ratio = \(2^2 : 3^2 = 4 : 9\).
15.
If \(a : b = c : d\), which mathematical property states that \((a + b) : (a - b) = (c + d) : (c - d)\)?
View Answer
Answer: Option A
Explanation:
Explanation:
Step 1: Given \(\frac{a}{b} = \frac{c}{d}\).
Step 2: Adding 1 to both sides gives \(\frac{a+b}{b} = \frac{c+d}{d}\) (Componendo).
Step 3: Subtracting 1 from both sides gives \(\frac{a-b}{b} = \frac{c-d}{d}\) (Dividendo).
Step 4: Dividing the first result by the second gives the property of Componendo and Dividendo.
Step 2: Adding 1 to both sides gives \(\frac{a+b}{b} = \frac{c+d}{d}\) (Componendo).
Step 3: Subtracting 1 from both sides gives \(\frac{a-b}{b} = \frac{c-d}{d}\) (Dividendo).
Step 4: Dividing the first result by the second gives the property of Componendo and Dividendo.
16.
In a 60L mixture, the ratio of milk to water is 2 : 1. How much water must be added to make the ratio 1 : 2?
View Answer
Answer: Option B
Explanation:
Explanation:
Step 1: Initial mixture components: Total parts = 3. Milk = \(\frac{2}{3} \times 60 = 40L\), Water = \(\frac{1}{3} \times 60 = 20L\).
Step 2: Let \(x\) be water added. New Water = \(20 + x\). Milk remains 40L.
Step 3: Set up new ratio: \(\frac{40}{20 + x} = \frac{1}{2}\).
Step 4: Cross-multiply: \(80 = 20 + x \implies x = 60L\).
Step 2: Let \(x\) be water added. New Water = \(20 + x\). Milk remains 40L.
Step 3: Set up new ratio: \(\frac{40}{20 + x} = \frac{1}{2}\).
Step 4: Cross-multiply: \(80 = 20 + x \implies x = 60L\).
17.
If \(x\) is subtracted from 14, 17, 34, and 42, the remaining numbers are in proportion. Find \(x\).
View Answer
Answer: Option B
Explanation:
Explanation:
Step 1: Set up the proportion: \(\frac{14 - x}{17 - x} = \frac{34 - x}{42 - x}\).
Step 2: Cross-multiply: \((14 - x)(42 - x) = (34 - x)(17 - x)\).
Step 3: Expand: \(588 - 56x + x^2 = 578 - 51x + x^2\).
Step 4: Simplify: \(588 - 56x = 578 - 51x \implies 10 = 5x \implies x = 2\).
Step 2: Cross-multiply: \((14 - x)(42 - x) = (34 - x)(17 - x)\).
Step 3: Expand: \(588 - 56x + x^2 = 578 - 51x + x^2\).
Step 4: Simplify: \(588 - 56x = 578 - 51x \implies 10 = 5x \implies x = 2\).
18.
The ratio of copper and zinc in Brass is 13 : 7. How much zinc is there in 100 kg of Brass?
View Answer
Answer: Option B
Explanation:
Explanation:
Step 1: Total parts = \(13 + 7 = 20\).
Step 2: Zinc share = \(\frac{7}{20}\) of the total mass.
Step 3: Calculation: \(\frac{7}{20} \times 100 = 35 kg\).
Step 2: Zinc share = \(\frac{7}{20}\) of the total mass.
Step 3: Calculation: \(\frac{7}{20} \times 100 = 35 kg\).
19.
An amount of money is distributed among A, B, C, and D in the ratio 3 : 4 : 9 : 10. If C gets $2,580 more than B, what is the total amount of A and D together?
View Answer
Answer: Option A
Explanation:
Explanation:
Step 1: Let the shares be \(3x, 4x, 9x, 10x\).
Step 2: Given \(C - B = 2580 \implies 9x - 4x = 2580\).
Step 3: Solve for \(x\): \(5x = 2580 \implies x = 516\).
Step 4: Total of A and D = \(3x + 10x = 13x\).
Step 5: Calculation: \(13 \times 516 = 6708\).
Step 2: Given \(C - B = 2580 \implies 9x - 4x = 2580\).
Step 3: Solve for \(x\): \(5x = 2580 \implies x = 516\).
Step 4: Total of A and D = \(3x + 10x = 13x\).
Step 5: Calculation: \(13 \times 516 = 6708\).
20.
If \(A : B = \frac{1}{2} : \frac{3}{8}\), \(B : C = \frac{1}{3} : \frac{5}{9}\), and \(C : D = \frac{5}{6} : \frac{3}{4}\), find the ratio \(A : B : C : D\).
View Answer
Answer: Option B
Explanation:
Explanation:
Step 1: Simplify individual ratios.
\(A : B = 4 : 3\)
\(B : C = 3 : 5\)
\(C : D = 10 : 9\)
Step 2: Combine A, B, and C: Since B is 3 in both, \(A : B : C = 4 : 3 : 5\).
Step 3: Link C to D: To make C common (from 5 to 10), multiply \(A:B:C\) by 2.
Step 4: Result: \(8 : 6 : 10 : 9\).
\(A : B = 4 : 3\)
\(B : C = 3 : 5\)
\(C : D = 10 : 9\)
Step 2: Combine A, B, and C: Since B is 3 in both, \(A : B : C = 4 : 3 : 5\).
Step 3: Link C to D: To make C common (from 5 to 10), multiply \(A:B:C\) by 2.
Step 4: Result: \(8 : 6 : 10 : 9\).