Question

one number, x, is 12 more than half another number, y. the sum of the numbers is 60. what are the numbers?
a. x = 24 and y = 36
b. x = 36 and y = 24
c. x = 32 and y = 28
d. x = 28 and y = 32

Explanation:

Step1: Translate the first condition

The first condition is "One number, \( x \), is 12 more than half another number, \( y \)". So we can write the equation as \( x=\frac{y}{2}+12 \).

Step2: Translate the second condition

The second condition is "The sum of the numbers is 60", so the equation is \( x + y=60 \).

Step3: Substitute and solve

Substitute \( x=\frac{y}{2}+12 \) into \( x + y=60 \). We get \( \frac{y}{2}+12 + y=60 \). Combine like terms: \( \frac{3y}{2}+12 = 60 \). Subtract 12 from both sides: \( \frac{3y}{2}=48 \). Multiply both sides by \( \frac{2}{3} \): \( y = 32 \). Then substitute \( y = 32 \) into \( x + y=60 \), we get \( x=60 - 32 = 28 \)? Wait, no, wait. Wait, maybe we can check the options directly. Let's check each option:

• Option A: \( x = 24 \), \( y = 36 \). Check \( x=\frac{y}{2}+12 \): \( \frac{36}{2}+12=18 + 12 = 30

eq24 \). So A is wrong.

• Option B: \( x = 36 \), \( y = 24 \). Check \( x=\frac{y}{2}+12 \): \( \frac{24}{2}+12 = 12 + 12 = 24

eq36 \). Wait, no, that's not right. Wait, maybe I made a mistake in the equation. Wait, the problem says "x is 12 more than half of y", so \( x=\frac{y}{2}+12 \). Let's check Option D: \( x = 28 \), \( y = 32 \). \( \frac{32}{2}+12=16 + 12 = 28 \). And \( 28 + 32 = 60 \). Oh! I see, earlier when I solved, I must have messed up the substitution. Let's do it again. From \( x + y=60 \), we have \( x = 60 - y \). Substitute into \( x=\frac{y}{2}+12 \): \( 60 - y=\frac{y}{2}+12 \). Subtract 12: \( 48 - y=\frac{y}{2} \). Add y to both sides: \( 48=\frac{3y}{2} \). Multiply by \( \frac{2}{3} \): \( y = 32 \). Then \( x = 60 - 32 = 28 \). So \( x = 28 \), \( y = 32 \), which is Option D. Wait, but let's check Option D: \( x = 28 \), \( y = 32 \). \( x=\frac{y}{2}+12 \): \( 16 + 12 = 28 \), correct. And \( x + y=60 \), correct. So the correct option is D. Wait, but earlier when I thought I made a mistake, but checking the options, D works.

Wait, maybe my initial substitution had an error. Let's re-express the first equation: \( x=\frac{y}{2}+12 \), so \( x - \frac{y}{2}=12 \). Second equation: \( x + y=60 \). Let's solve the system:

Multiply the first equation by 2: \( 2x - y = 24 \).

Second equation: \( x + y = 60 \).

Add the two equations: \( 3x = 84 \), so \( x = 28 \). Then \( y = 60 - 28 = 32 \). Yes, so \( x = 28 \), \( y = 32 \), which is Option D.

Answer:

D. \( x = 28 \) and \( y = 32 \)