Question

The larger of two numbers is 1 less than twice the smaller number. If 5 times the larger is equal to 3 times the smaller increased by 23, find the numbers.

A) Larger number: 10, Smaller number: 4
B) Larger number: 7, Smaller number: 3
C) Larger number: 8, Smaller number: 3
D) Larger number: 6, Smaller number: 2

Answer 1

After setting up two equations based on the given problem, we find that the smaller number is 4 and the larger number is 7, responding to the question with option A) Larger number: 7, Smaller number: 4.

Step-by-step explanation:

To solve the problem where the larger of two numbers is 1 less than twice the smaller number, let's designate the smaller number as x and the larger number as y. According to the problem, y = 2x - 1. Next, the statement that 5 times the larger is equal to 3 times the smaller increased by 23 translates to 5y = 3x + 23.

Now, let's substitute the first equation into the second:

1. 5(2x - 1) = 3x + 23
2. 10x - 5 = 3x + 23
3. 10x - 3x = 23 + 5
4. 7x = 28
5. x = 4

Now that we know x, the smaller number, is 4, we can find y by substituting x back into the equation y = 2x - 1:

1. y = 2(4) - 1
2. y = 8 - 1
3. y = 7

The larger number is 7 and the smaller number is 4. Therefore, the answer is option A) Larger number: 7, Smaller number: 4.