The ratio of two numbers A and B is 5 : 8. If 5 is added to each of A and B, then the ratio of A and B becomes 2 : 3. The sum of A and B is:
65
Solving Ratio Problems: Finding the Sum of Two Numbers
Let's break down this ratio problem step-by-step to find the sum of the two numbers A and B.
Understanding the Initial Ratio of A and B
We are given that the initial ratio of two numbers, A and B, is 5 : 8. This means that for some common factor 'x', we can represent the numbers as:
- A = 5x
- B = 8x
Setting up the Equation After Adding to A and B
The problem states that if 5 is added to both A and B, their new ratio becomes 2 : 3.
- New A = A + 5 = 5x + 5
- New B = B + 5 = 8x + 5
The ratio of these new numbers is given as 2 : 3. We can write this as an equation:
\(\frac{\text{New A}}{\text{New B}} = \frac{2}{3}\)
Substituting the expressions for New A and New B, we get:
\(\frac{5x + 5}{8x + 5} = \frac{2}{3}\)
Solving for the Common Factor 'x'
To solve for 'x', we can cross-multiply:
\(3 \times (5x + 5) = 2 \times (8x + 5)\)
Distribute the numbers on both sides:
\(15x + 15 = 16x + 10\)
Now, we need to isolate 'x'. Subtract \(15x\) from both sides:
\(15 = 16x - 15x + 10\)
\(15 = x + 10\)
Subtract 10 from both sides:
\(15 - 10 = x\)
\(5 = x\)
So, the common factor 'x' is 5.
Finding the Original Numbers A and B
Now that we have the value of 'x', we can find the original numbers A and B:
- A = 5x = \(5 \times 5 = 25\)
- B = 8x = \(8 \times 5 = 40\)
Let's check if the ratio after adding 5 is correct:
- New A = 25 + 5 = 30
- New B = 40 + 5 = 45
The new ratio is \(30 : 45\). Dividing both by their greatest common divisor, 15, we get \(30 \div 15 : 45 \div 15 = 2 : 3\). This matches the given information, so our value for 'x' is correct.
Calculating the Sum of A and B
The question asks for the sum of A and B. We found A = 25 and B = 40.
Sum = A + B = \(25 + 40 = 65\)
Summary of Steps
Here is a quick summary of how we solved the problem:
- Represent A and B using the initial ratio and a variable (x).
- Set up an equation using the new ratio after adding 5 to A and B.
- Solve the equation for the variable x.
- Calculate the values of A and B using the found value of x.
- Calculate the sum of A and B.
| Step | Description | Calculation/Representation |
|---|---|---|
| 1 | Initial Ratio Representation | A = 5x, B = 8x |
| 2 | Ratio After Adding 5 | \(\frac{5x + 5}{8x + 5} = \frac{2}{3}\) |
| 3 | Solve for x | \(3(5x+5) = 2(8x+5)\) \(15x + 15 = 16x + 10\) \(x = 5\) |
| 4 | Find A and B | A = \(5 \times 5 = 25\) B = \(8 \times 5 = 40\) |
| 5 | Calculate Sum | Sum = \(25 + 40 = 65\) |
The sum of A and B is 65.
Revision Table: Ratio and Proportion Concepts
| Concept | Explanation | Example |
|---|---|---|
| Ratio | A comparison of two quantities by division. Represented as a:b or a/b. | If there are 3 apples and 5 bananas, the ratio of apples to bananas is 3:5. |
| Proportion | An equation stating that two ratios are equal. | \(\frac{a}{b} = \frac{c}{d}\) is a proportion. |
| Cross-Multiplication | Method used to solve equations involving proportions: if \(\frac{a}{b} = \frac{c}{d}\), then \(ad = bc\). | Solving \(\frac{x}{4} = \frac{3}{6}\) gives \(6x = 4 \times 3 \implies 6x = 12 \implies x = 2\). |
Additional Information: Applying Ratios
Ratio problems often appear in various contexts, such as mixing ingredients, scaling maps, comparing speeds, and sharing quantities. Understanding how to represent numbers based on their ratio and setting up proportional equations is key to solving these problems.
- When a value is added to or subtracted from quantities in a ratio, the relationship changes, requiring a new equation based on the new ratio.
- Always use a variable (like 'x') to represent the common multiplier in the initial ratio to maintain the relationship between the numbers.
- Verify your answer by plugging the calculated numbers back into the conditions given in the problem.
Important Questions from Simple Ratios
The ratio of two numbers A and B is 5: 8. If 5 is added to each of A and B, then the ratio becomes 2 : 3. The difference between A and B is:
A sum of Rs. 6342 is divided amongst A, B, C and D in the ratio 3 : 4 : 8 : 6. What is the difference between the shares of B and D?
The ratio of monthly incomes of A and B is 4 ∶ 5 and that of their monthly expenditures is 3 ∶ 8. If the income of A is equal to the expenditure of B, then what is the ratio of savings of A and B?
The ratio of the monthly incomes of A and B is 11 : 13 and the ratio of their expenditures is 9 : 11. If both of them manage to save Rs. 4,000 per month, then find the difference in their incomes (in Rs.)
The train ticket fare from places A to B in 2 nd class AC and 3 rd class AC is Rs. 2,500 and Rs. 2,000, respectively. If the fares of 2 nd class AC and 3 rd class AC are increased by 20% and 10%, respectively, then find the ratio of the new fares of 2 nd class AC and 3 rd class AC.
