What is the compounded ratio of (2 ∶ 5), (5 ∶ 11) and (33 ∶ 8)?
3 ∶ 4
Understanding Compounded Ratio
The compounded ratio of two or more ratios is obtained by multiplying together the antecedents (first terms) of all the ratios and also multiplying together the consequents (second terms) of all the ratios. The resulting ratio is the product of the antecedents to the product of the consequents.
Calculating the Compounded Ratio
We are given three ratios:
- Ratio 1: 2 ∶ 5 or $\frac{2}{5}$
- Ratio 2: 5 ∶ 11 or $\frac{5}{11}$
- Ratio 3: 33 ∶ 8 or $\frac{33}{8}$
To find the compounded ratio, we multiply the antecedents together and the consequents together.
Product of antecedents = $2 \times 5 \times 33$
Product of consequents = $5 \times 11 \times 8$
The compounded ratio is $(2 \times 5 \times 33) : (5 \times 11 \times 8)$.
We can write this as a fraction and simplify:
$$ \text{Compounded Ratio} = \frac{\text{Product of Antecedents}}{\text{Product of Consequents}} = \frac{2 \times 5 \times 33}{5 \times 11 \times 8} $$
Now, we simplify the fraction by cancelling common factors from the numerator and the denominator:
$$ \frac{2 \times 5 \times 33}{5 \times 11 \times 8} = \frac{2 \times 5 \times (3 \times 11)}{5 \times 11 \times 8} $$
Cancel out 5 from the numerator and denominator:
$$ \frac{2 \times (3 \times 11)}{11 \times 8} $$
Cancel out 11 from the numerator and denominator:
$$ \frac{2 \times 3}{8} $$
Now, simplify $\frac{2}{8}$ which is $\frac{1}{4}$:
$$ \frac{1 \times 3}{4} = \frac{3}{4} $$
So, the simplified compounded ratio is 3 : 4.
Resulting Compounded Ratio
The compounded ratio of (2 ∶ 5), (5 ∶ 11), and (33 ∶ 8) is 3 ∶ 4.
Revision Table: Key Concepts in Ratios
| Term | Explanation | Example |
|---|---|---|
| Ratio | A comparison of two quantities by division. Represented as a : b or a/b. | 3 : 4 |
| Antecedent | The first term of a ratio (a in a : b). | In 3 : 4, the antecedent is 3. |
| Consequent | The second term of a ratio (b in a : b). | In 3 : 4, the consequent is 4. |
| Compounded Ratio | The ratio obtained by multiplying the antecedents and consequents of two or more ratios. | Compounded ratio of (a:b) and (c:d) is (ac : bd). |
Additional Information on Ratios and Proportions
Ratios are fundamental in mathematics and are used to express relationships between quantities. They are often used in areas like scaling, mixing, comparing values, and in solving problems involving proportions.
- Inverse Ratio: The inverse ratio of a : b is b : a.
- Duplicate Ratio: The duplicate ratio of a : b is a² : b².
- Sub-duplicate Ratio: The sub-duplicate ratio of a : b is √a : √b.
- Triplicate Ratio: The triplicate ratio of a : b is a³ : b³.
- Sub-triplicate Ratio: The sub-triplicate ratio of a : b is ³√a : ³√b.
- Proportion: An equality of two ratios. If a : b = c : d, then ad = bc (product of extremes equals product of means).
Understanding how to compound ratios is useful when dealing with problems involving successive ratios or combining multiple ratio relationships.
Important Questions from Compound Ratios
The sum of three numbers is 280. If the ratio between the first and second numbers is 2 : 3 and the ratio between second and third numbers is 4 : 5, find the second number.
When x is subtracted from each of the numbers 54, 49, 22 and 21, the numbers so obtained are in proportion. The ratio of (8x - 25) to (7x - 26) is:
The train fare, bus fare and air fare between 2 places are in the ratio 5 : 8 : 12, the number of passenger travelled by them is in the ratio 3 : 4 : 5 and the total fare collected on a particular day for these modes of transportation for a single trip is Rs. 1,07,000. Find the fare collected from the air passengers.
A person carries Rs. 165/ - in the form of currency notes of denominations Rs. 5, Rs. 10 & Rs. 20 in the ratio of 3 : 2 : 1. What is the value of currency notes of Rs. 20 denomination?
If a: b = 5: 3, then (a³-b³): (a³+b³) = ?
