Question

Elliott went to the store for a bag of cookies. When he left the store he gave his mom half the cookies and half a cookie. He then came upon his aunt and gave her half the remaining cookies and half a cookie. Continuing on his way, he found his friend and gave him half the cookies left and half a cookie. When made it to class, he only had six cookies left. How many did he start out with?

Answer 1

40 cookies

Explanation:

Let x be the number of cookies Elliott started with.

After giving his mom half, he had x/2 + 1 cookie left.

After giving his aunt half, he had x/4 + 1 cookie left.

After giving his friend half, he had x/8 + 1 cookie left.

Since he had 6 cookies left, then x/8 + 1 = 6

x/8 = 5

x = 40 cookies.

So the answer is 40

Answer 2

55 cookies

Explanation:

Let's start with building the equation.

1. Let's say that Elliott's starting amount is an unknown number
`x`

He gave his mom half the cookies and half a cookie

This means he has
`x` x
`(1)/(2)` - 0.5

He gave his aunt half of the remaining cookies and half a cookie

This means he has (
`x` x
`(1)/(2)` - 0.5 ) x
`(1)/(2)` - 0.5

He gave his friend half the remaining cookies and half a cookie

This means he has ((
`x` x
`(1)/(2)` - 0.5 ) x
`(1)/(2)` - 0.5) x
`(1)/(2)` - 0.5

So, when Elliott made it to class, the equation above equaled to 6 cookies.

((
`x` x
`(1)/(2)` - 0.5 ) x
`(1)/(2)` - 0.5) x
`(1)/(2)` - 0.5 = 6

2. We will now traverse this equation:

((
`x` x
`(1)/(2)` - 0.5 ) x
`(1)/(2)` - 0.5) x
`(1)/(2)` - 0.5 = 6

Add 0.5 (half a cookie) to each side of the equation

((
`x` x
`(1)/(2)` - 0.5 ) x
`(1)/(2)` - 0.5) x
`(1)/(2)` - 0.5 (+ 0.5 )= 6 (+ 0.5)

((
`x` x
`(1)/(2)` - 0.5 ) x
`(1)/(2)` - 0.5) x
`(1)/(2)` = 6.5

Multiply 2 on each side of the equation

((
`x` x
`(1)/(2)` - 0.5 ) x
`(1)/(2)` - 0.5) x
`(1)/(2)` (x 2)= 6.5 (x 2)

(
`x` x
`(1)/(2)` - 0.5 ) x
`(1)/(2)` - 0.5 = 13

Add 0.5 (half a cookie) to each side of the equation again

(
`x` x
`(1)/(2)` - 0.5 ) x
`(1)/(2)` - 0.5 (+ 0.5 ) = 13(+ 0.5 )

((
`x` x
`(1)/(2)` - 0.5 ) x
`(1)/(2)` - 0.5) x
`(1)/(2)` = 13.5

Multiply 2 on each side of the equation

((
`x` x
`(1)/(2)` - 0.5 ) x
`(1)/(2)` (x 2)= 13.5 (x 2)

`x` x
`(1)/(2)` - 0.5 = 27

Add 0.5 (half a cookie) to each side of the equation again

`x` x
`(1)/(2)` - 0.5 (+ 0.5 ) = 27 (+ 0.5 )

`x` x
`(1)/(2)` = 27.5

Multiply 2 on each side of the equation

`x` x
`(1)/(2)` (x 2) = 27.5 (x 2)

`x` = 55