Question

the sum of two numbers is 22. three times one number increased by 5 is the same as twice the other number decreased by 4. what is the larger of the two numbers?
15
11
7
16
clear my selection

Explanation:

Step1: Define variables

Let the two numbers be \( x \) and \( y \). We know that \( x + y = 22 \), so \( y = 22 - x \).

Step2: Set up the second equation

Three times one number increased by 5 is \( 3x + 5 \), and twice the other number decreased by 4 is \( 2y - 4 \). Since they are equal, we have the equation:
\( 3x + 5 = 2y - 4 \)

Step3: Substitute \( y \)

Substitute \( y = 22 - x \) into the second equation:
\( 3x + 5 = 2(22 - x) - 4 \)

Step4: Simplify the equation

First, expand the right side: \( 3x + 5 = 44 - 2x - 4 \)
Simplify the right side: \( 3x + 5 = 40 - 2x \)
Add \( 2x \) to both sides: \( 3x + 2x + 5 = 40 \)
Simplify: \( 5x + 5 = 40 \)
Subtract 5 from both sides: \( 5x = 40 - 5 \)
\( 5x = 35 \)
Divide both sides by 5: \( x = \frac{35}{5} = 7 \)

Step5: Find the other number

Since \( x + y = 22 \) and \( x = 7 \), then \( y = 22 - 7 = 15 \)

Step6: Determine the larger number

Compare \( x = 7 \) and \( y = 15 \). The larger number is 15.

Answer:

15