Solving Ratio Problems Involving Linear Equations

Ratio problems are a common type of algebraic problem that often involve setting up equations. Here, we explore different examples and solve them step-by-step. These methods are useful for improving your problem-solving skills in mathematics.

Example 1: Identifying Two Numbers from a Ratio 2:3 to 3:4

Problem: If eight is added to both numbers in the ratio 2:3, the new ratio becomes 3:4. What are the original numbers?

Solution: Let's denote the two numbers as 2x and 3x. According to the problem, when eight is added to both numbers, the new ratio is 3:4. This gives us the equation:

frac{2x 8}{3x 8} frac{3}{4}

Cross-multiplying to eliminate the fraction, we get:

4(2x 8) 3(3x 8)

Expanding both sides:

8x 32 9x 24

Now, let's rearrange the equation to solve for x:

32 - 24 9x - 8x

8 x

Once we have the value of x, we can find the original numbers:

The first number is: 2x 2(8) 16 The second number is: 3x 3(8) 24

To verify the solution, add 8 to both 16 and 24:

16 8 24 24 8 32

Now, check the new ratio:

32 / 24 4 / 3.5 3 / 4

This confirms that our solution is correct. The original numbers are 16 and 24.

Example 2: Another Ratio Problem

Problem: Let the numbers be 3a and 4a, and when 8 is added to both numbers, the new ratio becomes 3:4. Determine the numbers.

Solution: Given the ratio 3a 8 / 4a 8 4 / 5, we start by cross-multiplying:

5(3a 8) 4(4a 8)

Expanding both sides:

15a 40 16a 32

Rearranging to solve for a:

40 - 32 16a - 15a

8 a

Therefore, the numbers are 24 and 32.

Example 3: Ratio Involving Addition of 4

Problem: The ratio of two numbers is 2:3. When 4 is added to both numbers, the new ratio becomes 3:4. What are the original numbers?

Solution: Let the two numbers be 2k and 3k. When 4 is added to both numbers:

2k 4 : 3k 4 3 : 4

We can write:

3(3k 4) 4(2k 4)

Expanding both sides:

9k 12 8k 16

Rearranging to solve for k:

9k - 8k 16 - 12

k 4

Thus, the original numbers are:

2k 2(4) 8 3k 3(4) 12

Verification: Adding 4 to both numbers:

8 4 12 12 4 16

Check the new ratio:

12 / 16 3 / 4

Our solution is correct. The original numbers are 8 and 12.

Example 4: Ratio Involving Substitution and Equations

Problem: Let two numbers be x and y. If x/y 2/3, and (x 4) / (y 4) 3/4, find the numbers.

Solution: Starting with the given ratios:

x 2/3 y

From the second equation:

(x 4) / (y 4) 3/4

Substituting 2y/3 for x:

((2y/3) 4) / (y 4) 3/4

Clearing the fraction:

4((2y 12)/3) 3(y 4)

Expanding and simplifying:

8y 48 9y 36

8y - 9y 36 - 48

-y -12

y 12

x 2/3 * 12 8

Therefore, the two numbers are 8 and 12.

Conclusion

Solving problems involving ratios and linear equations is a fundamental skill in algebra. By following the methods demonstrated in these examples, you can systematically tackle similar problems with confidence and accuracy.

Keywords

ratio, linear equations, problem solving