Sure, let's solve the problem step-by-step.
We are given two pieces of information:
1. A is greater than B by 1/3 of the sum of A and B.
2. If B is increased by 40, it becomes greater than twice A by 10.
Let's denote [tex]\( A \)[/tex] and [tex]\( B \)[/tex] as the unknown values we need to find. We can set up the following equations based on the given information:
1. From the first statement, we have:
[tex]\[ A = B + \frac{1}{3}(A + B) \][/tex]
2. From the second statement, we have:
[tex]\[ B + 40 = 2A + 10 \][/tex]
We will solve these equations step-by-step.
### Step 1: Solve the first equation for [tex]\( A \)[/tex] in terms of [tex]\( B \)[/tex]
From the first statement:
[tex]\[ A = B + \frac{1}{3}(A + B) \][/tex]
Let's clear the fraction by multiplying through by 3:
[tex]\[ 3A = 3B + A + B \][/tex]
Combine the like terms:
[tex]\[ 3A = 4B + A \][/tex]
Now subtract [tex]\( A \)[/tex] from both sides to isolate [tex]\( A \)[/tex]:
[tex]\[ 2A = 4B \][/tex]
Divide both sides by 2:
[tex]\[ A = 2B \][/tex]
### Step 2: Substitute [tex]\( A = 2B \)[/tex] into the second equation
We have:
[tex]\[ B + 40 = 2A + 10 \][/tex]
Substitute [tex]\( A = 2B \)[/tex] into this equation:
[tex]\[ B + 40 = 2(2B) + 10 \][/tex]
Simplify:
[tex]\[ B + 40 = 4B + 10 \][/tex]
Subtract [tex]\( B \)[/tex] from both sides:
[tex]\[ 40 = 3B + 10 \][/tex]
Subtract 10 from both sides:
[tex]\[ 30 = 3B \][/tex]
Divide both sides by 3:
[tex]\[ B = 10 \][/tex]
### Step 3: Find [tex]\( A \)[/tex] using [tex]\( B \)[/tex]
We found that [tex]\( B = 10 \)[/tex]. Now use [tex]\( A = 2B \)[/tex]:
[tex]\[ A = 2 \cdot 10 = 20 \][/tex]
### Conclusion
Thus, the values of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are [tex]\( A = 20 \)[/tex] and [tex]\( B = 10 \)[/tex]. Therefore, the correct answer is:
(C) 20, 10
