Nathan had 4/7 as many pens as Henry at first. After Nathan bought 19 more pens and Henry gave away 8 pens, they had an equal number of pens in the end. How many pens did Nathan have at first?
Henry had x
Nathan had 4x/7
Nathan got 19 more ----> now has 4x/7 + 19
Henry gave away 8 ---- now has x - 8
4x/7 + 19 = x - 8
4x/7 - x = -27
4x - 7x = -189
x = 63
Nathan had 4x/7 at first
= (4/7)(63) = 36
Let's break down the problem step-by-step:
Step 1: Let's assume the number of pens Henry had at first as x.
Step 2: Nathan had 4/7 as many pens as Henry, so Nathan had (4/7)*x pens at first.
Step 3: After Nathan bought 19 more pens, he had [(4/7)*x] + 19 pens.
Step 4: Henry gave away 8 pens, so he had x - 8 pens.
Step 5: In the end, both Nathan and Henry had an equal number of pens. Therefore, we can set up the equation:
[(4/7)*x] + 19 = x - 8
Step 6: Simplify the equation by multiplying both sides by 7 to remove the fraction:
4x + 133 = 7x - 56
Step 7: Combine like terms:
-3x = -189
Step 8: Solve for x by dividing both sides by -3:
x = (-189) / (-3) = 63
Step 9: Now, substitute the value of x into the expression for Nathan's pens at first:
Nathan had (4/7)*x = (4/7)*63 = 36 pens at first.
Therefore, Nathan had 36 pens at first.
To solve this problem, let's break it down step by step:
Step 1: Write down the given information.
- Nathan had 4/7 as many pens as Henry at first.
- After Nathan bought 19 more pens, and Henry gave away 8 pens, they had an equal number of pens in the end.
Step 2: Assign variables to the unknowns.
Let's assume that the number of pens Henry had at first is H.
So, Nathan had 4/7 * H pens at first.
Step 3: Find the total number of pens after the changes.
After Nathan bought 19 more pens, he had 4/7 * H + 19 pens.
After Henry gave away 8 pens, he had H - 8 pens.
Step 4: Set up an equation.
Since they had an equal number of pens in the end, we can set up the following equation:
4/7 * H + 19 = H - 8
Step 5: Solve the equation.
First, we can simplify the equation by getting rid of the fraction by multiplying both sides by 7:
4H + 133 = 7H - 56
Next, let's isolate the H variable on one side of the equation. We can do this by subtracting 4H from both sides and adding 56 to both sides:
133 + 56 = 7H - 4H
189 = 3H
Finally, divide both sides by 3 to solve for H:
189 / 3 = H
H = 63
Step 6: Find Nathan's initial number of pens.
Since Nathan had 4/7 as many pens as Henry, we can calculate Nathan's initial number of pens:
Nathan's initial number of pens = 4/7 * Henry's initial number of pens
Nathan's initial number of pens = 4/7 * 63
Nathan's initial number of pens = 36
Therefore, Nathan had 36 pens at first.