In this explainer, we will learn how to estimate quotients of fractions and mixed numbers by rounding them to the nearest half or whole.

We can use estimation to give us a general idea of the answer to a calculation. Estimation can be very useful in situations such as casual planning. Redecorating a room, for example, may require an estimation of how many tins of paint or rolls of wallpaper to use. Estimation will seldom give us the correct answer, if we need an exact value, we would use the specific values to work out a calculation exactly. For those situations where an estimate of fractions is sufficient or required, let us now look at how we can do this.

### How To: Estimating a Calculation Involving Fractions

We can use one, or both, of the methods below to help us with estimating a calculation involving fractions:

- Round the fraction to the nearest half (0, , or 1).
- This can be helpful for simpler fractions or when the problem is not particularly complex. For example, we can say that an estimation for is .

- Simplify the fractions using factors to help.
- For example, is close to which can be simplified to .
- is close to which can be simplified to .

Let us now look at some examples of how we can use estimation of fractions to find the sums, products, differences, and quotients of fractions in a calculation.

### Example 1: Estimating the Sum of Fractions

Two children are making a cake. One has cups of flour and the other has cups of flour. Estimate how many cups of flour they have together by rounding to the nearest half.

### Answer

Considering the fraction , we can check if the fraction component, , is closer to 0 or by a visual diagram.

Here, we can see that the fraction is closer to than it is to 0.

Alternatively, we can also compare these by considering each as a decimal, which does not require exact drawing skills like the previous diagram.

Recall that can be written as the recurring decimal , which can be written as .

The fraction can be written as 0.5.

We can see that since our fraction is at a distance of from 0, but only a distance of from , it is therefore closer to than 0 and we can say .

So, we can write

Next, to consider an estimation for the fraction , we can consider if is closer to 0, , or 1.

It can be helpful to recall that since and , we know that will be between and 1. As the numerator 5 is closer to 4 than 8, we can estimate

So, for our fraction, , we estimate

Therefore, we have

This gives us the answer that the children have an estimated 8 cups of flour.

### Example 2: Estimating the Quotient of Fractions

Estimate by rounding to the nearest half.

### Answer

To estimate , we consider whether this is closer to 0, , or 1.

If we split an object into 6 equal pieces, then and . Therefore, since the 5 in the fraction is closer to 6 than 3, we can say that

Similarly, to find an estimate to the nearest half for the fraction , we can recall that and . Since 7 in the fraction is closer to 8 than 4, we have

Therefore, we can write

So, our answer is that an estimate for is 1.

### Example 3: Estimating the Difference and Product of Fractions

If we estimate fractions by rounding to the nearest half, which of the following pairs of fractions have an estimated difference of and an estimated product of ?

### Answer

Let us consider the pair .

Since is closer to than , this means that

Since is closer to than 0, we can write

To test the difference of our estimations for and , we would have

The product of our estimations for and would give

The pair of fractions that give an estimated difference and product of is , .

To check the other pairs of fractions, we consider their estimated values by rounding to the nearest half and we evaluate the difference and product of these estimated values. We can see the results in the table below.

Pair | Estimated Values | Estimated Difference | Estimated Product |
---|---|---|---|

Since there is only one pair of fractions that have an estimated difference of and an estimated product of , the answer is

### Example 4: Estimating the Quotient of Fractions

12 pounds of oatmeal are going to be packaged in bags that hold each. By rounding to whole numbers, estimate the number of bags that will be used.

### Answer

To round the fraction to the nearest whole number, we can consider that if we have 9 parts, then half of that would be 4.5 parts. Since we have 5 parts out of 9 in our fraction, , we have more than half and so this part of the fraction would round up to 1.

This will give us that

So, to calculate the number of bags of oatmeal, we need to divide 12 by the estimate for . This will give us

So our estimated answer is that 3 bags will be used.

In this topic, it is worth noting that it is possible to use different numbers or fractions for an estimation, and both estimations can be equally valid. Let us look at an example of estimation with two possible values for the estimation result.

### Example 5: Estimating the Sum of Fractions

Estimate to the nearest half.

### Answer

We can use estimation to the nearest half to help round our fractions. We can say that and

Therefore, we can estimate as

### Key Points

- We can use estimation of fractions by rounding the fraction part of a mixed number to the nearest half. This will give us a good estimation if there are simple fractions.
- When estimating fractions to the nearest whole or half, it is helpful to be adept with finding equivalent fractions, particularly fractions equivalent to .
- If we have a fraction without a whole number constituent, it is usually best to avoid rounding this to 0, particularly when dividing and the divisor would be 0, which would give a mathematical error.