Explainer: Estimating Quotients of Fractions

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 Estimate a Calculation Involving Fractions

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

  1. Round the fraction to the nearest half (0, 12, 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 78 is 88=1.
  2. Simplify the fractions using factors to help.
    • For example, 516 is close to 416 which can be simplified to 14.
    • 41118 is close to 41218 which can be simplified to 423.

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 313 cups of flour and the other has 458 cups of flour. Estimate how many cups of flour they have together by rounding to the nearest half.

Answer

Considering the fraction 313, we can check if the fraction component, 13, is closer to 0 or 12 by a visual diagram.

Here, we can see that the fraction 13 is closer to 12 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 13 can be written as the recurring decimal 0.333, which can be written as 0.3.

The fraction 12 can be written as 0.5.

We can see that since our fraction 13 is at a distance of 0.3 from 0, but only a distance of 0.16 from 12, it is therefore closer to 12 than 0 and we can say 1312.

So, we can write 313312.

Next, to consider an estimation for the fraction 458, we can consider if 58 is closer to 0, 12, or 1.

It can be helpful to recall that since 48=12 and 88=1, we know that 58 will be between 12 and 1. As the numerator 5 is closer to 4 than 8, we can estimate 5812.

So, for our fraction, 458, we estimate 458412.

Therefore, we have 313+458312+4128.

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

Example 2: Estimating the Quotient of Fractions

Estimate 56÷78 by rounding to the nearest half.

Answer

To estimate 56, we consider whether this is closer to 0, 12, or 1.

If we split an object into 6 equal pieces, then 36=12 and 66=1. Therefore, since the 5 in the fraction 56 is closer to 6 than 3, we can say that 56661.

Similarly, to find an estimate to the nearest half for the fraction 78, we can recall that 48=12 and 88=1. Since 7 in the fraction 78 is closer to 8 than 4, we have 78881.

Therefore, we can write 56÷781÷11.

So, our answer is that an estimate for 56÷78 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 12 and an estimated product of 12?

  1. 911,49
  2. 13,16
  3. 110,13
  4. 18,67
  5. 1011,110

Answer

Let us consider the pair 911,49.

Since 911 is closer to 1111 than 5.511(=12), this means that 91111111.

Since 49 is closer to 12(=4.59) than 0, we can write 4912.

To test the difference of our estimations for 911 and 49, we would have 9114911212.

The product of our estimations for 911 and 49 would give 911×491×1212.

The pair of fractions that give an estimated difference and product of 12 is 911, 49.

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.

PairEstimated ValuesEstimated DifferenceEstimated Product
13,1612,0120=1212×0=0
110,130,12120=120×12=0
18,670,110=10×1=0
1011,1101,010=11×0=0

Since there is only one pair of fractions that have an estimated difference of 12 and an estimated product of 12, the answer is 911,49.

Example 4: Estimating the Quotient of Fractions

12 pounds of oatmeal are going to be packaged in bags that hold 359pounds each. By rounding to whole numbers, estimate the number of bags that will be used.

Answer

To round the fraction 59 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, 59, we have more than half and so this part of the fraction would round up to 1.

This will give us that 3594.

So, to calculate the number of bags of oatmeal, we need to divide 12 by the estimate for 359. This will give us 12÷35912÷43.

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 249+345 to the nearest half.

Answer

We can use estimation to the nearest half to help round our fractions. We can say that 249212 and 3454.

Therefore, we can estimate 249+345 as 249+345212+4612.

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 12.
  • 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.

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