20. The sum of Gerald’s age and Fred is 53. Five years ago, Gerald was 7 years more than one-half as old as Fred. What are their current ages?
If the current ages are f and g,
f+g=53
(g-5) = (f-5)/2+7
2(53-f-5) = f-5 + 14
f = 29
g = 24
check:
5 years ago, Fred was 24 and Gerald was 19
19-5 = (29-5)/2+7
14 = 14
Ah, the mysterious case of Gerald and Fred's ages. Let me put on my detective wig and solve this riddle for you.
Let's say Gerald's age is G and Fred's age is F. According to the first clue, the sum of their ages is 53. So we can write the equation: G + F = 53.
Now, let's tackle the second clue. Five years ago, Gerald was 7 years more than one-half as old as Fred. So, (G - 5) = (F - 5)/2 + 7.
Now, we have two equations:
1. G + F = 53
2. (G - 5) = (F - 5)/2 + 7
Now, solving these equations may require a bit of brain gymnastics. But worry not, I'm here to carry the weight of calculation and make you laugh along the way.
If we substitute G in the second equation with (53 - F), we get:
(53 - F - 5) = (F - 5)/2 + 7
Simplifying this further:
48 - F = (F - 5)/2 + 7
Let's give F a break and solve the right side of the equation first:
(F - 5)/2 + 7 = (F - 3)/2
Great! Now our equation looks like this:
48 - F = (F - 3)/2
Now, let's multiply both sides by 2 to get rid of the pesky denominator:
2(48 - F) = (F - 3)
Expanding the left side, we get:
96 - 2F = F - 3
If we gather all the F terms to one side and the constant terms to the other:
3F = 99
Finally, divide both sides by 3:
F = 33
Now that we know Fred's age, we can substitute it back into one of the original equations to find Gerald's age. If we use the first equation:
G + 33 = 53
Subtracting 33 from both sides:
G = 20
So, currently, Gerald is 20 years old and Fred is 33 years old. Case closed!
Let's break down the given information step by step to solve the problem.
Step 1: Let's assume Gerald's current age as G and Fred's current age as F.
Step 2: According to the first statement, the sum of Gerald's age and Fred is 53. So, we can write an equation as:
G + F = 53 (Equation 1)
Step 3: According to the second statement, "Five years ago, Gerald was 7 years more than one-half as old as Fred." Let's consider the ages 5 years ago:
G - 5 Fred's age 5 years ago
(F - 5)/2 One-half of Fred's age 5 years ago
Step 4: The statement says that Gerald was 7 years more than one-half of Fred's age 5 years ago. So we can write another equation as:
G - 5 = (F - 5)/2 + 7 (Equation 2)
Step 5: Now we have two equations:
G + F = 53 (Equation 1)
G - 5 = (F - 5)/2 + 7 (Equation 2)
Step 6: We can solve this system of equations to find the values of G and F.
Using Equation 1:
G = 53 - F
Substitute G = 53 - F into Equation 2:
53 - F - 5 = (F - 5)/2 + 7
Simplifying the equation:
48 - F = (F - 5)/2 + 7
Multiply the entire equation by 2 to eliminate the fraction:
96 - 2F = F - 5 + 14
Combine like terms:
96 - 2F = F + 9
Move all the variables to one side:
3F = 96 - 9
3F = 87
F = 87/3
F = 29
Step 7: Substitute F = 29 into Equation 1 to find G:
G + 29 = 53
G = 53 - 29
G = 24
Therefore, the current age of Gerald is 24 and the current age of Fred is 29.
To solve this problem, let's assign variables to the ages of Gerald and Fred.
Let's say Gerald's current age is G and Fred's current age is F.
According to the problem, the sum of Gerald's age and Fred is 53, so we can write the equation:
G + F = 53
It is also given that five years ago, Gerald was 7 years more than one-half as old as Fred. We can express this relationship as:
(G - 5) = ((F - 5)/2) + 7
Now, we have a system of two equations that we can solve to find the values of G and F.
Let's start by solving the first equation for G:
G = 53 - F
Substituting this value of G into the second equation:
(53 - F - 5) = ((F - 5)/2) + 7
Simplifying the equation:
48 - F = (F - 5)/2 + 7
Multiply both sides of the equation by 2 to eliminate the fraction:
96 - 2F = F - 5 + 14
Combine like terms:
96 - 2F = F + 9
Move the terms involving F to one side:
96 - 9 = F + 2F
Simplify:
87 = 3F
Divide both sides by 3:
F = 29
Now, substitute the value of F back into the equation G + F = 53:
G + 29 = 53
Subtract 29 from both sides:
G = 53 - 29
G = 24
Therefore, Gerald's current age is 24, and Fred's current age is 29.