Question

Mrs Chen bought some eggs. She used 1/2 of them to make tarts and 1/4 of the remainder to make a cake. She had 9 eggs left. How many eggs did she buy?

Answer 1

She starts with 24 eggs.

Explanation:

We can work through this question, by using only fractions.

Mrs Chen used `1/2` of the eggs.

This means she has `1/2`of the eggs left.

She uses another `1/4` of the `1/2` that are left.
`1/4 xx 1/2 = 1/8`

Together she has used:

`1/2+1/8`

`= 4/8 +1/8 = 5/8`

If `5/8` have been used, it means that `8/8-5/8 = 3/8` is left.

`3/8` of the total number of eggs is `9` eggs

`1/8` of the total is `9 div 3 = 3` eggs

`8/8` is the total number of eggs. `3xx 8 = 24`eggs

Check:
`1/2 xx 24 = 12` eggs used. 12 eggs are left

`1/4 xx 12 = 3` more eggs are used.

`12-3 = 9` eggs are left from the 24.

Answer 2

I have chosen to write this in a similar way to that used by algebra as an introduction to the methods.

`"total count" = 24" eggs"`

Explanation:

First use `->1/2`
Second use `->1/4 " remainder "->1/4xx1/2=1/8`

Total use is:

`color(blue)(1/2+1/8)color(red)(" " ->" " [1/2xx4/4]+1/8)color(green)(" " =" " 4/8+1/8)color(purple)(" "=" "5/8)`

Therefor not used is: `" "1-5/8 = 3/8" total count"`

But not used is 9 eggs

`=>color(brown)( 3/8" total count "=" "9" eggs")`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
But we need 1 of total count. Not `3/8` of it. Thus we need to change the `3/8` into 1

Multiply both sides by `color(blue)(8/3)`
Note that this is where the shortcut method of cross multiply comes from.

`color(brown)(color(blue)(8/3xx)[3/8xx" total count"] = color(blue)(8/3xx)[9" eggs"])`

`3/3xx8/8xx"total count"=(8xx9)/3" eggs"`

`" "1xx1xx"total count"=(72)/3" eggs"`

Note that `1xx"total count is the same as total count"`

`color(white)()`

`" "color(purple)(bar(ul(|color(white)(2/2)"total count" = 24" eggs"color(white)(2/2)|)))`