**Further estimating to find the operator** *KS3 Revision*

## What you need to know

**Things to remember:**

- We can use estimation to simplify fractions.
- Try each of our four operations, then compare the answers.
- Try all four operations might not be necessary.

*Find the missing operation that makes the following true:*

* *

5 \_\_\_ 2\frac{3}{4} = 7.75

This might be obvious, but let’s try all four operations to see what we get.

**Addition: ** 5 + 2\dfrac{3}{4} = 7.75

**Subtraction: ** 5 - 2\dfrac{3}{4} =2.25

**Multiplication: ** 5 \times 2\dfrac{3}{4} = 13.75

**Division: ** 5 \div 2\dfrac{3}{4} =1.\overline{81}

** **

So, we can see that our missing operation here must have been a +.

We don’t always have to try every operation though, if we remember some key points:

**Addition:** When adding a positive number to another number, it makes it bigger.

**Subtraction:** When subtracting a positive number from another number, it makes it smaller.

**Multiplication: **When multiplying by a number bigger than 1, it makes it bigger.

When multiplying by a number smaller than 1, it makes it smaller.

**Division:** When dividing by a number bigger than 1, it makes it smaller.

When dividing by a number smaller than 1, it makes it bigger.

The numbers and fraction were quite nice, but how about if they are a bit tougher?

*Find the missing operation that makes the following true:*

* *

\dfrac{13}{48} \_\_\_ 3 = \frac{13}{16}

Neither of these fractions are particularly nice, but we can estimate find estimate for them to make them a little nicer. We can estimate fractions by changing the numerator and denominator into numbers that let us simplify them down.

** **

\dfrac{13}{48}\rightarrow\frac{12}{48}\rightarrow\frac{1}{4}

\dfrac{13}{16}\rightarrow\frac{12}{16}\rightarrow\frac{3}{4}

** **

Now that we have our estimates, let’s put them in.

\dfrac{1}{4} \_\_\_ 3 = \frac{3}{4}

Our estimate is now asking “How do we use 3 to turn \dfrac{1}{4} into \dfrac{3}{4}?” Well, we multiply of course!

\dfrac{1}{4} \times 3 = \frac{3}{4}

So, our missing operation was a multiplication!

\dfrac{13}{48} \times 3= \frac{13}{16}

## Example Questions

**Question 1:** Find the missing operation that makes the following true:

\frac{21}{100} \_\_\_ 2 = \frac{21}{50}

\dfrac{21}{100}\rightarrow\frac{20}{100}\rightarrow\dfrac{1}{5}

\dfrac{21}{50}\rightarrow\frac{20}{50}\rightarrow\dfrac{2}{5}

Our estimate is now asking “How do we use 2 to turn \dfrac{1}{5} into \dfrac{2}{5}?” Well, we multiply, so this must be our missing operation!

\dfrac{21}{100} \times 2 = \dfrac{21}{50}

**Question 2:** Find the missing operation that makes the following true:

* *

\frac{9}{40} \_\_\_ 4 = \frac{9}{10}

\dfrac{9}{40}\rightarrow\dfrac{10}{40}\rightarrow\dfrac{1}{4}

\dfrac{9}{10}\rightarrow\dfrac{10}{10}\rightarrow 1

Our estimate is now asking “How do we use 4 to turn \dfrac{1}{4} into 1 whole?” Well, we multiply, so this must be our missing operation!

\dfrac{9}{40} \times 4 = \dfrac{9}{10}