Solving Ratio and LCM Problems: A Comprehensive Guide

Understanding the relationship between ratio and least common multiple (LCM) is essential for solving a variety of mathematical problems. This guide will walk you through solving problems involving ratio and LCM, providing detailed steps and examples for clarity.

Example 1: Ratio 3:5 and LCM 225

Consider two numbers in the ratio 3:5, and their LCM is 225. Let these numbers be represented as 3x and 5x.

Step 1: Set up the equation for LCM.
LCM(3x, 5x) 15x

Step 2: Use the given LCM to find x.
15x 225

Step 3: Solve for x.
x 225 / 15 15

Step 4: Determine the smaller number.
Smaller number 3x 3 × 15 45

Example 2: Handling Common Factors

The first method may not work if the numbers share a common factor. In such cases, we need to use the second method.

Step 1: Let the numbers be a and b, with HCF z.
xyz LCM(a, b)

Step 2: Substitute the given values.
3×4×z 240 12z

Step 3: Solve for z.
z 240 / 12 20

Step 4: Determine the numbers.
x 3, y 4

Final numbers: 3×20 60 and 4×20 80

Example 3: Ratio 4:5 with LCM 180

Suppose two numbers are in the ratio 4:5, and their LCM is 180.

Step 1: Represent the numbers as 4x and 5x.

Step 2: Set up the equation for LCM.

Step 3: Use the given LCM to find x.
2 180

Step 4: Solve for x.
x 180 / 20 9

Step 5: Determine the numbers.
4x 4 × 9 36

5x 5 × 9 45

Solving with Algebraic Equations

Consider x and y representing two numbers, with x/y 4/5 and LCM(x, y) 180.

Step 1: Utilize the ratio to express one variable in terms of the other.
x/y 4/5

Step 2: Set up the equation for LCM.
xy 180

Step 3: Substitute and solve for x.
x 4y/5

Step 4: Solve the system of equations.
4y/5 × 5x/4 180

Step 5: Simplify and solve for x.
5x^2 / 4 180

Step 6: Solve for x^2 and then for x.
5x^2 720

Step 7: Use the positive solution for x (absolute value considerations for negative solutions).
x^2 144

Step 8: Determine the values of x and y.

x 12 or -12

When x 12:
x/y 4/5 → 12/y 4/5

Step 9: Solve for y.
y/12 5/4

Step 10: Substitute the value of x to find y.
y 5 x 12 / 4 15

When x -12:
x/y 4/5 → -12/y 4/5

Step 11: Solve for y.
y/-12 5/4

Step 12: Substitute the value of x to find y.
y 5 x -12 / 4 -15

The two sets of numbers are thus (12, 15) and (-12, -15).

However, the LCM of 12 and 15 is 60. Multiply by three to get the correct LCM of 180.

Final numbers: 36 and 45 (or -36 and -45).

These examples illustrate the step-by-step process of solving ratio and LCM problems, ensuring a comprehensive understanding of the topic for students and educators.