Question

What is `5/8` as a decimal?

Answer 1

`color(magenta)("Method 1 of 2")`

0.625

Explanation:

You would have been told the shortcut way of just dividing 8 into 5. Easy if you have a calculator. Sometimes not so if you have to do it manually.

What follows is just one method out of several.

There some fractions that it is advisable that you commit their decimal equivalents to memory. This is one of them.

Known: `1/8=0.125`

So `5/8" "=" "5xx0.125`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Rather than worry about the decimal when you multiply think of `0.125`; consider it as `125/(color(red)(1000)` or the same thing `125xx1/(color(red)(1000)`. They all have the same intrinsic value.

`color(blue)("Step 1"`
`color(brown)("Just for a moment do not think about the "xx1/color(red)(1000))`
Write as:
`" "125`
`" "ul(color(white)(...)5) larr" Multiply"`
`" "625`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 2"`

`color(brown)(" Now we must think about the "xx1/color(red)(1000))" "` giving:

`" "125`
`" "ul(color(white)(...)5) larr" Multiply"`
`" "625 " "-> 625xx1/(color(red)(1000))=0.625`

Answer 2

Another look at the eighths....

Explanation:

The halves, quarters and eighths are all part of the same family and you can combine them to make fraction work easier.

Compare these fractions with their percents and decimals.

"Half of" is the same as `div 2`

`1 div 2 = color(blue)(1/2) " "100% div 2 = color(blue)(50%) " " 1/2 = color(blue)(0.5)`

`1/2 div 2 = color(teal)(1/4)" "50% div 2 = color(teal)(25%) " "1/4 = color(teal)(0.25)`

`1/4 div 2 = color(orange)(1/8) " "25% div 2 = color(orange)(12 1/2%) " "1/8 = color(orange)(0.125)`

"Counting in eighths" would go like this
Compare them with the decimals and percents and find the pattern:

`1/8,color(white)(xxx)1/4, color(white)(xxx)3/8, color(white)(xxx)1/2, color(white)(xxx)color(red)(5/8), color(white)(xxx)3/4, color(white)(xxx)7/8 color(white)(xxx)8/8`

`0.125" "0.25 " "0.375" "0.5" "color(red)(0.625)" "0.75" "0.875" "1.00"`

`12 1/2" "25" "37 1/2" "50" " color(red)(62 1/2)" "75" " 87 1/2" " 100%`

It is VERY good idea to learn these heart. Here is one example

"Find `62 1/2% " of " 560` imported cars"

`color(red)(62 1/2%) xx 560/1 " easily becomes " cancel(62 1/2%) color(red)(5/8) xx 560/1`

`= 5/cancel8 xxcancel560^70/1 = 350 " cars"`

Answer 3

`color(magenta)("Method 2 of 2")`

Explanation:

`color(blue)("What follows is just another way of writing long devision")`

This approach accumulates adjustments which are then applied at the end. You will understand as I go along

8 is more than 5 so we change the 5 to `50 xx1/10`
The `1/10` is the adjustment. Every time we need to divide 8 into a part of the number (digit) that is less than 8 we introduce another adjusted value.

Starting point `5` is the same as `50xx1/10`

`color(white)("ddddddd")50color(green)(xx1/10)`
`color(magenta)(6)xx8->ul(48larr" Subtract"`
`ul(color(white)("dddddddd")2larr" Change this to 20")`
`color(white)("ddddddd")20color(green)(xx1/10)`
`color(magenta)(2)xx8->ul(16larr" Subtract")`
`ul(color(white)("dddddddd")4 larr" Change this to 40")`
`color(white)("ddddddd")40color(green)(xx1/10)`
`color(magenta)(5)xx8->ul(40larr" Subtract")`
`color(white)("dddddddd")0larr" As we have zero we stop"`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We have accumulated the adjustments of `color(green)(1/10xx1/10xx1/10)`

The number we have is `color(magenta)(625)`

Putting it all together:

`color(magenta)(625)color(green)(xx1/10xx1/10xx1/10)`

`color(magenta)(625)color(green)(xx1/1000)color(white)("d")=color(white)("d")color(blue)(0.625)`