Solving Ratio Problems: A Comprehensive Guide

Understanding and solving ratio problems can be a daunting task for many, but with the right approach, these types of problems can be as simple as adding numbers. This article provides a detailed guide to solving a specific type of ratio problem, offering step-by-step solutions and explanations that can help enhance your mathematical skills.

Understanding the Problem

We are given a number problem where two numbers are in a given ratio. Initially, we have two numbers in the ratio 4:7. When each of these numbers is increased by 4, the new ratio becomes 3:5. The task is to find the larger number.

Step-by-Step Solution

Method 1: Using Variables and Equations

Step 1: Let the two numbers be x and y, where x : y 4 : 7. This implies:

[ frac{x}{y} frac{4}{7} rightarrow y frac{7}{4}x ]

Step 2: When each number is increased by 4, the new ratio becomes 3:5. Therefore:

[ frac{x 4}{y 4} frac{3}{5} rightarrow 5(x 4) 3(y 4) ]

Substituting y frac{7}{4}x from (mathJax:

[ 5(x 4) 3left(frac{7}{4}x 4right) ]

Simplifying this:

[ 5x 20 frac{21}{4}x 12 ]

Multiplying throughout by 4 to clear the fraction:

[ 2 80 21x 48 ]

Subtracting 2 from both sides:

[ 80 x 48 ]

Therefore:

[ x 32 ]

Substituting back for y:

[ y frac{7}{4}(32) 56 ]

Thus, the larger number is:

y 56

Method 2: Using Algebraic Substitution

Step 1: Let each number be expressed as a multiple of a variable. Suppose the numbers are 4x and 7x. The initial ratio is 4:7.

Step 2: When each of these numbers is increased by 4, the new ratio becomes 3:5. This gives the equation:

[ frac{4x 4}{7x 4} frac{3}{5} ]

Cross-multiplying to solve for x:

[ 5(4x 4) 3(7x 4) ]

Simplifying:

[ 2 20 21x 12 ]

Subtracting 2 from both sides:

[ 20 x 12 ]

Therefore:

[ x 8 ]

The larger number is:

7x 7(8) 56

Method 3: System of Equations

Step 1: Let the two numbers be a and b with the initial ratio 3:5. When each number is increased by 10, the new ratio becomes 5:7.

This gives us the system of equations:

[ frac{3x 10}{5x 10} frac{5}{7} ]

Cross-multiplying to solve for x:

[ 7(3x 10) 5(5x 10) ]

Simplifying:

[ 21x 70 25x 50 ]

Subtracting 21x 50 from both sides:

[ 20 4x ]

Therefore:

[ x 5 ]

The larger number is:

5x 5(5) 25

Conclusion

By applying various methods, we can effectively solve ratio problems. The larger number in the given problem is 56. These methods not only provide the final answer but also offer insight into the underlying mathematical principles, helping to strengthen your problem-solving skills in mathematics.

Keywords

ratio problem, increasing ratio, solving equations