Solving Ratio Problems Involving Subtracting a Constant

Ratio problems that involve subtracting a constant from each term to achieve a new ratio are common in mathematics. This article will explore techniques to solve these types of problems using equations, algebraic manipulation, and cross multiplication.

Understanding the Problem

Consider the problem: Two numbers are in the ratio 3:5. If 9 is subtracted from the numbers, the ratio becomes 12:23. What are the numbers? This problem involves a change in the ratio of two numbers after a certain constant is subtracted from each of them. We will break this down step-by-step.

Solution to the Example Problem

Let the numbers be 3x and 5x where x is the common multiple.

After subtracting 9 from each number, the new numbers are 3x-9 and 5x-9. The new ratio is given as 12:23. Therefore, we can write the equation:

[ frac{3x - 9}{5x - 9} frac{12}{23} ]

Cross-multiplying to eliminate the fractions:

[ 23(3x - 9) 12(5x - 9) ]

Expanding and simplifying the equation:

[ 69x - 207 6 - 108 ]

Isolating the variable:

[ 69x - 6 207 - 108 ]

Simplifying further:

[ 9x 99 ]

Solving for x:

[ x 11 ]

Substituting x back into the original expressions:

[ 3x 3(11) 33 ]

[ 5x 5(11) 55 ]

Conclusion

Therefore, the original numbers are 33 and 55. If 9 is subtracted from each, the new numbers are 24 and 46, resulting in the ratio 24:46, which simplifies to 12:23, as required by the problem.

Practice Problems

1. Two numbers are in the ratio 3:5. If 7 is subtracted from each, the new numbers are in the ratio 12:23. What is the smaller number?

2. Two numbers are in the ratio 4:7. If 11 is subtracted from each, the new numbers are in the ratio 17:31. What are the original numbers?

3. Two numbers are in the ratio 5:9. If 13 is subtracted from each, the new numbers are in the ratio 14:25. What is the smaller number?

Key Takeaways

Identify the common multiple (x) and express the original numbers as 3x and 5x. Write the equation based on the given ratio after the subtraction. Use cross multiplication to simplify and solve for the variable (x). Substitute the value of x back to find the original numbers.

Mastering these steps will help in solving a variety of ratio problems involving the subtraction of a constant.