If a : b : c = \(\frac{1}{4} : \frac{1}{3} : \frac{1}{2}, \) then \( \ \frac{a}{b} : \frac{b}{c} : \frac{c}{a} = ?\)
9 : 8 : 24
Understanding Ratio Problems with Fractions
This question involves working with ratios, specifically converting a ratio with fractions into a simpler integer ratio and then using that ratio to find another related ratio. Let's break down the steps involved to solve this ratio problem effectively.
Step 1: Convert the Fractional Ratio a:b:c to an Integer Ratio
The given ratio is \(a : b : c = \frac{1}{4} : \frac{1}{3} : \frac{1}{2}\). To convert this fractional ratio into a simple integer ratio, we need to find the least common multiple (LCM) of the denominators of the fractions (4, 3, and 2).
- The denominators are 4, 3, and 2.
- The LCM of 4, 3, and 2 is 12.
Now, multiply each term in the ratio by the LCM (12):
- \(a = \frac{1}{4} \times 12 = 3\)
- \(b = \frac{1}{3} \times 12 = 4\)
- \(c = \frac{1}{2} \times 12 = 6\)
So, the integer ratio \(a : b : c\) is \(3 : 4 : 6\).
Step 2: Calculate the Individual Ratios \( \frac{a}{b}, \frac{b}{c}, \frac{c}{a} \)
Using the integer values we found for a, b, and c (which are proportional to the original fractional values), we can calculate the required ratios:
- \( \frac{a}{b} = \frac{3}{4} \)
- \( \frac{b}{c} = \frac{4}{6} = \frac{2}{3} \) (simplifying the fraction)
- \( \frac{c}{a} = \frac{6}{3} = 2 = \frac{2}{1} \) (simplifying and writing as a fraction)
Step 3: Form the Ratio \( \frac{a}{b} : \frac{b}{c} : \frac{c}{a} \) and Convert to an Integer Ratio
The ratio we need to find is \( \frac{a}{b} : \frac{b}{c} : \frac{c}{a} \), which is \( \frac{3}{4} : \frac{2}{3} : \frac{2}{1} \). Again, we have a ratio with fractions. To convert this to an integer ratio, we find the LCM of the denominators (4, 3, and 1).
- The denominators are 4, 3, and 1.
- The LCM of 4, 3, and 1 is 12.
Multiply each term in the ratio \( \frac{3}{4} : \frac{2}{3} : \frac{2}{1} \) by the LCM (12):
- First term: \( \frac{3}{4} \times 12 = 3 \times 3 = 9 \)
- Second term: \( \frac{2}{3} \times 12 = 2 \times 4 = 8 \)
- Third term: \( \frac{2}{1} \times 12 = 2 \times 12 = 24 \)
So, the integer ratio \( \frac{a}{b} : \frac{b}{c} : \frac{c}{a} \) is \(9 : 8 : 24\).
Summary of Calculations
| Original Ratio | LCM of Denominators | Integer Ratio a:b:c |
|---|---|---|
| \( \frac{1}{4} : \frac{1}{3} : \frac{1}{2} \) | 12 | 3 : 4 : 6 |
| Component Ratio | Calculation | Value |
|---|---|---|
| \( \frac{a}{b} \) | \( \frac{3}{4} \) | \( \frac{3}{4} \) |
| \( \frac{b}{c} \) | \( \frac{4}{6} \) | \( \frac{2}{3} \) |
| \( \frac{c}{a} \) | \( \frac{6}{3} \) | \( \frac{2}{1} \) |
| Ratio of Components | LCM of Denominators | Final Integer Ratio |
|---|---|---|
| \( \frac{3}{4} : \frac{2}{3} : \frac{2}{1} \) | 12 | 9 : 8 : 24 |
The final calculated ratio \( \frac{a}{b} : \frac{b}{c} : \frac{c}{a} \) is \(9 : 8 : 24\). This corresponds to one of the given options.
Revision Table: Key Ratio Concepts
| Concept | Explanation | Example |
|---|---|---|
| Ratio | A comparison of two or more quantities. | \(a : b\) or \( \frac{a}{b} \) |
| Converting Fractional Ratio to Integer Ratio | Multiply each term by the LCM of the denominators. | \( \frac{1}{2} : \frac{1}{3} \implies (\frac{1}{2} \times 6) : (\frac{1}{3} \times 6) \implies 3 : 2 \) |
| Simplifying Ratios | Divide all terms by their greatest common divisor (GCD). | \(10 : 15 : 25 \implies 2 : 3 : 5 \) (Dividing by 5) |
Additional Information: Properties of Ratios
- Ratios can be simplified by dividing all parts by the same non-zero number.
- Ratios can be converted into fractions or percentages.
- If \(a : b = x : y\), then \( \frac{a}{b} = \frac{x}{y} \).
- Compound ratio of \(a:b\) and \(c:d\) is \((ac):(bd)\).
- Duplicate ratio of \(a:b\) is \(a^2:b^2\).
- Sub-duplicate ratio of \(a:b\) is \( \sqrt{a}:\sqrt{b} \).
- Triplicate ratio of \(a:b\) is \(a^3:b^3\).
- Sub-triplicate ratio of \(a:b\) is \( \sqrt[3]{a}:\sqrt[3]{b} \).
Important Questions from Ratio and Proportion
The cost of a diamond is directly proportional to the square of its weight. The cost of a 14 gm diamond is Rs. 2560. This diamond got broken down into two pieces in the ratio of 5 ∶ 9. How much loss percent is incurred due to this breakage ? (Correct to two decimal places)
Atul purchased Bread costing Rs.20 and gave a 100 rupee note to the shopkeeper. The shopkeeper gave the balance money in coins of denomination Rs.2, Rs.5 and Rs.10. If these coins are in the ratio 5 ∶ 4 ∶ 1, then how many Rs.5 coins did the shopkeeper give?
A person divides a certain amount among his three sons in the ratio of 3 ∶ 4 ∶ 5. If he had divided this amount in the ratio of 1/3,1/4,1/5, his son, who had got the lowest share earlier, would get Rs.1,188 more. Find the amount (in Rs).
In a school 3/8 of the number of students are girls and the rest are boys. One-third of the number of boys are below 10 years and 2/3 the number if girls are also below 10 years. If the number of students of age 10 or more years is 260. then the number of boys in the school is:
A, B and C divide a certain sum of money among themselves. The average of the amount with them is Rs.4520. Share of A is \(10\frac{2}{3}%\) % more than share of B and \(33\frac{1}{3}%\) % less than share of C. What is the share of B (in Rs.)?
