The sum of two positive numbers p and q is 32, while their difference is 12. What is the smaller number?

• A. 12
• B. 10
• C. 14
• D. 8

Answer: (B)

To solve for the smaller number given that the sum of two numbers ( p + q = 32 ) and their difference ( p - q = 12 ), we can use a system of equations to find the values of ( p ) and ( q ).

First, we can express the equations as follows:

1. ( p + q = 32 )

2. ( p - q = 12 )

To eliminate one variable, we can add these two equations together. When we do this, the ( q ) terms will cancel out:

[

(p + q) + (p - q) = 32 + 12

]

This simplifies to:

[

2p = 44

]

Then, we can divide both sides by 2 to find ( p ):

[

p = 22

]

Now that we have ( p ), we can substitute this value back into one of the original equations to find ( q ). Using the first equation:

[

22 + q = 32

]

Subtracting 22 from both sides gives us:

[

q = 10

]

Since we found that ( p = 22 ) and ( q