The compounded ratio of (1 ∶ 3), (6 ∶ 5) and (7 ∶10) is:
7/25
Understanding ratios is fundamental in mathematics. A ratio compares two quantities. When we have multiple ratios, we can combine them to form a single compounded ratio.
Calculating the Compounded Ratio
The question asks for the compounded ratio of (1 ∶ 3), (6 ∶ 5), and (7 ∶ 10).
To find the compounded ratio of several ratios, we multiply the antecedents (the first terms) together and multiply the consequents (the second terms) together.
Given ratios are:
- 1 ∶ 3 (or $1/3$)
- 6 ∶ 5 (or $6/5$)
- 7 ∶ 10 (or $7/10$)
Let's identify the antecedents and consequents:
- Antecedents: 1, 6, 7
- Consequents: 3, 5, 10
Now, we calculate the product of the antecedents:
Product of Antecedents $= 1 \times 6 \times 7$
Product of Antecedents $= 42$
Next, we calculate the product of the consequents:
Product of Consequents $= 3 \times 5 \times 10$
Product of Consequents $= 15 \times 10$
Product of Consequents $= 150$
The compounded ratio is the ratio of the product of the antecedents to the product of the consequents.
Compounded Ratio $= \text{Product of Antecedents} \ratio; \text{Product of Consequents}$
Compounded Ratio $= 42 \ratio; 150$
We can express this ratio as a fraction and simplify it:
Compounded Ratio $= \frac{42}{150}$
To simplify the fraction, we find the greatest common divisor (GCD) of 42 and 150. Both numbers are divisible by 2 and by 3. So, they are divisible by $2 \times 3 = 6$.
Divide both the numerator and the denominator by 6:
Numerator $= 42 \div 6 = 7$
Denominator $= 150 \div 6 = 25$
The simplified compounded ratio is $7/25$.
Therefore, the compounded ratio of (1 ∶ 3), (6 ∶ 5) and (7 ∶10) is 7 ∶ 25 or $7/25$.
| Ratio | Antecedent | Consequent |
|---|---|---|
| 1 ∶ 3 | 1 | 3 |
| 6 ∶ 5 | 6 | 5 |
| 7 ∶ 10 | 7 | 10 |
| Calculation Step | Result |
|---|---|
| Product of Antecedents | $1 \times 6 \times 7 = 42$ |
| Product of Consequents | $3 \times 5 \times 10 = 150$ |
| Compounded Ratio (unsimplified) | 42 ∶ 150 or $42/150$ |
| Compounded Ratio (simplified by dividing by 6) | 7 ∶ 25 or $7/25$ |
Revision Table: Types of Ratios
| Type of Ratio | Description | Example |
|---|---|---|
| Simple Ratio | Compares two quantities. | a ∶ b |
| Duplicate Ratio | Square of a ratio. | Duplicate ratio of a ∶ b is $a^2$ ∶ $b^2$ |
| Sub-duplicate Ratio | Square root of a ratio. | Sub-duplicate ratio of a ∶ b is $\sqrt{a}$ ∶ $\sqrt{b}$ |
| Triplicate Ratio | Cube of a ratio. | Triplicate ratio of a ∶ b is $a^3$ ∶ $b^3$ |
| Sub-triplicate Ratio | Cube root of a ratio. | Sub-triplicate ratio of a ∶ b is $\sqrt[3]{a}$ ∶ $\sqrt[3]{b}$ |
| Inverse Ratio (Reciprocal Ratio) | Ratio with antecedent and consequent interchanged. | Inverse ratio of a ∶ b is b ∶ a |
| Compounded Ratio | Product of antecedents ∶ Product of consequents of given ratios. | Compounded ratio of (a ∶ b) and (c ∶ d) is (ac ∶ bd) |
Additional Information on Ratios
Ratios are used to show how much of one quantity there is compared to another quantity. They can be written in different forms:
- Using the symbol ∶ (e.g., 1 ∶ 3)
- As a fraction (e.g., $1/3$)
- Using the word "to" (e.g., 1 to 3)
It's important that the quantities being compared in a ratio have the same units. When finding a compounded ratio, we essentially multiply fractions together. The concept of compounded ratios is useful in various applications, such as calculating combined probabilities or dealing with proportions in finance and other fields.
Simplifying a ratio means dividing both terms by their greatest common divisor (GCD) to get the ratio in its simplest form, just like simplifying a fraction.
Important Questions from Compound Ratios
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.
Rs. 750 are divided among A, B and C in such a manner that A : B = 5 : 2 and B : C = 7 : 13, What is A’s share?
Rs. 750 are divided among A, B and C in such a manner that A : B = 5 : 2 and B : C = 7 : 13, What is B’s share?
Two numbers are in the ratio 3 : 5. If 4 is subtracted from each of these two numbers, the new numbers then are in the ratio 4 : 7. Find the smaller of the two original numbers.
A person has some coins of Rs. 10, Rs. 5, and Rs. 2 denominations. The ratio of the products of the numbers of Rs. 10 and Rs. 5 coins, the numbers of Rs. 5 and Rs. 2 coins, and the numbers of Rs. 2 and Rs. 10 coins is 3 ∶ 4 ∶ 2 respectively. What could be the minimum amount of money this person has?
