Gavin has 106 pieces of $5 notes and $10 notes. He used 2/3 of the $10 notes and put in another six $5 notes. As a result, the number of $5 notes was fives times the number of $10 notes. What was the total value of the $5 notes Gavin have at first?

Question

Gavin has 106 pieces of $5 notes and $10 notes. He used 2/3 of the $10 notes and put in another six $5 notes. As a result, the number of $5 notes was fives times the number of $10 notes. What was the total value of the $5 notes Gavin have at first?

Answer 1

if there are f 5's and t 10's, then we know that

f+t = 106
f+6 = 5(1/3 t)
f=64 and t=42
check: 5*14 = 6+64✅
so the 42 $5 notes were worth $210

Answer 2

As x pieces of $5

y pieces of 10 x + y = 106 =>
y = 106 - x (1)
1/3y * 5 = x + 6 (2)
Put (1) into (2)
5/3 (106 - x) = x + 6
5 (106 - x) = 3x + 18
530 - 5x = 3x + 18
8x = 530 - 18 x = 512/8 = 64
64 * 5 = 320

Answer 3

my typo - my bad.

Way to watch, Anonymous.

Answer 4

To solve this problem, let's break it down into smaller steps.

Let's assume that Gavin initially had x $5 notes and y $10 notes.

1. We know that Gavin initially had a total of 106 notes, so we can write the equation x + y = 106.

2. We also know that after Gavin removes 2/3 of his $10 notes, he puts in an additional 6 $5 notes. This means that the number of $5 notes becomes 5 times the number of $10 notes. Mathematically, we can represent this as x + 6 = 5(y - (2/3)y).

Let's solve these two equations simultaneously to find the values of x and y.

Equation 1: x + y = 106
Equation 2: x + 6 = 5(y - (2/3)y)

To simplify Equation 2, we can distribute the 5:

x + 6 = 5y - (10/3)y

Combining like terms:

x + 6 = (15/3 - 10/3)y
x + 6 = (5/3)y

Now, let's solve for y by isolating y in Equation 2:

(5/3)y = x + 6

Multiply both sides of the equation by 3/5 to isolate y:

y = (3/5)(x + 6)

Now, substitute the expression for y in Equation 1:

x + (3/5)(x + 6) = 106

Multiply through by 5 to eliminate the fraction:

5x + 3(x + 6) = 530
5x + 3x + 18 = 530
8x + 18 = 530
8x = 530 - 18
8x = 512
x = 512/8
x = 64

Now that we have the value of x, we can substitute it back into Equation 1 to find y:

64 + y = 106
y = 106 - 64
y = 42

So, Gavin initially had 64 $5 notes and 42 $10 notes.

To find the total value of the $5 notes Gavin initially had, we multiply the number of $5 notes by the denomination:

Total value = 64 * $5 = $320.

Therefore, the total value of the $5 notes Gavin initially had is $320.