Algebra 1 word problem

Question

A cashier has 26 bills consisting of three times as many ones as fives, one less ten than fives, and the rest twenties. If the value is $120, find how many of each she has.

Answer 1

f represents the number of fives

3f represents the nyumber of ones

f - 1 represents the number of tens

26 - (f + 3f + (f - 1)) represents the number of twenty-fives

5f + 3f + 10(f - 1) + 25(26 - (f + 3f + (f - 1)) = 120

Can you solve for f and answer?

Answer 2

Hi Ho K

There are couple of approaches to consider, it is given that Fives, Ones, and Tens can be represented in terms of their relationship to Fives.

I will let

x = number of 5’s

3x = number of 1’s

x -1 = number of 10’s

Lots of amounts in terms of x

On the other hand while the Twenties can be represented by the difference of the total minus the other bills it can also be represented by another variable like y

y = number 20’s

We are given the total numbers of bills 26 and their total value $120.00

For the Total number of Bills we can use Equation 1

x + 3x + x -1 + y = 26

Combining like terms

5x -1 + y =26

We get the constants on one side of the equation by adding 1 to both sides

5x + y = 27

y = 27 - 5x

For the total costs we multiply the number of bills by their corresponding values

5x + 3x +10(x - 1) + 20y = 120

Multiply to remove the parentheses, then combine like terms

5x + 3x + 10x - 10 + 20y = 120

18x -10 + 20y = 120

Again get constants on one side of the equation by adding 10 to both sides and we have equation 2

18x + 20y = 130

Now we have two equations in two unknowns and we can solve by elimination

5x + y = 27

18x + 20y = 130

I will multiply equation 1 by -20 then add it to equation 2 to eliminate y

-100x - 20y = -540

18x +20y = 130

This gives

-82x = -410

x= -410/-82 = 5

y = 27 - 5(5) = 27 - 25 = 2

Checking

Substituting x and y in our previous set of equations

x + 3x + (x - 1) + y = 26

5 + 3(5) + (5 - 1) + 2

5 + 15 + 4 + 2 = 26 Total Bills

Value

5x + 3x + 10(x -1) + 20y = 120

5(5) + 3(5) + 10(4) + 20(2)

25 + 15 + 40 + 40 = 120

Hope this helps