Numbers Finder by Ratio and LCM

Input the ratio and LCM of two numbers to instantly determine their actual values.

Calculate Two Numbers Using Their Ratio and LCM

Result

How to Calculate Two Numbers Using Their Ratio and LCM

Suppose the two numbers are \( x \) and \( y \), and their ratio is \( r_1 : r_2 \), with the least common multiple \( \text{LCM} \) given.

Steps

1. Understand the ratio

Assume the two numbers can be expressed as: \( x = k \cdot r_1, \quad y = k \cdot r_2 \) Here, \( k \) is a proportional coefficient. The ratio between the numbers is \( r_1 : r_2 \).

2. Determine the LCM

Since \( \text{LCM}(x, y) = k \cdot \text{LCM}(r_1, r_2) \), the proportional coefficient \( k \) can be calculated as: \( k = \frac{\text{LCM}}{\text{LCM}(r_1, r_2)} \)

3. Find \( x \) and \( y \)

Substitute \( k \) into the expressions for \( x \) and \( y \): \( x = k \times r_1 \) \( y = k \times r_2 \)

Examples

Example 1: The ratio of two numbers is 1:4 and their LCM is 280. Find the two numbers.

Solution:

1. Compute the LCM:

\( \text{LCM}(1, 4) = 4 \)

2. Compute the proportional coefficient \( k \):

\( k = \frac{280}{4} = 70 \)

3. Substitute \( k \) into the formulas for \( x \) and \( y \):

\( x = 70 \times 1 = 70 \)

\( y = 70 \times 4 = 280 \)

Result: The two numbers are 70 and 280.

Example 2: The ratio of two numbers is 13:21 and their LCM is 4641. Find the two numbers.

Solution:

1. Compute the LCM:

\( \text{LCM}(13, 21) = 273 \)

2. Compute the proportional coefficient \( k \):

\( k = \frac{4641}{273} = 17 \)

3. Substitute \( k \) into the formulas for \( x \) and \( y \):

\( x = 17 \times 13 = 221 \)

\( y = 17 \times 21 = 357 \)

Result: The two numbers are 221 and 357.