Question

One of the toughest races in the world is the 26.2 mile Boston Marathon. After running 22 miles, Sheila is exhausted. What percent of the race must she still run?

Answer 1

`16 4/131%` Exact answer

`16.03%` to 2 decdfimal places

Explanation:

`color(blue)("Preamble")`

Percentage is just a fraction but a special one where the bottom number (denominator) is fixed at 100

Some times you see percentage presented as a number followed by the symbol %. The thing is; you can think of the % as another way of writing `xx1/100`

Example: `60% ->60xx% -> 60xx1/100=60/100`
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`color(blue)("Answering the question")`

The whole race is over 26.2 miles. So everything is a fraction of this.

She ran 22 miles so as a fraction of the whole this is `22/26.2`

What was left as a fraction of the whole was `(26.2-22)/26.2 = 4.2/26.2`

But this is not in the form of percentage.

As a fraction what we need is: `("some number")/100`

`color(brown)("Shortcut method")`

`4.2-:26.2 xx 100` and stick a % on the end giving:

`-> 16.03%" to 2 decimal places"` by calculator
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`color(brown)("First principle meth")`

Starting point `4.2/26.2` don't like decimals so lets get rid of them. Multiply by 1 but in the form of `10/10`

`4.2/26.2xx10/10=42/262` check this with a calculator they have
`color(white)("dddddddddddddddd")`the same value.

We need to change the bottom number to 100. Note that `262xx100/262=100`. For multiply or divide what you do to the top you do to the bottom

`(42xx100/262)/(262xx100/262) =( 16 4/131) /100 = 16 4/131% larr" Exact answer"`

As a decimal `16.03%" to 2 decimal places "larr" Not exact"`