 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

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

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

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!

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

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

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

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!

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