What is a Back-to-Back Stem-and-Leaf Plot?

A back-to-back stem-and-leaf plot is used to compare two data distributions. The ‘stem’ is shared by both sets of data which are listed in the ‘leaves’ to the left and right of the stem respectively. The data is ordered so that the larger values are further from the stem.

The back-to-back stem-and-leaf plot below shows the data comparing the test scores of two classes.

In this example, the stem tells us the tens digit of each number and the numbers in the leaves tell us the ones digit.

For example, the 7 shown in line with the 2 in the stem represents 27.

The 5 shown in line with the 1 stem represents 15.

On a back-to-back stem and leaf plot, a key is drawn to help explain how to read the diagram. The key on a stem-and-leaf plot illustrates an example.

For example, 7 |1| means 17. The digit of 7 is in the left leaf and so, it is read backwards.

|1| 2 means 12. Since the 2 is in the right leaf, the number is read forwards.

How to Draw a Back-to-Back Stem-and-Leaf Diagram

To make a back-to-back stem-and-leaf diagram:

1. Put the numbers in each data set in order from smallest to largest.
2. All apart from the final digit of each number is listed so that they appear in the central stem once.
3. The final digits in data set 1 are listed to the left of the stem, getting larger as they get further from the stem.
4. The final digits in data set 2 are listed to the right of the stem, getting larger as they get further from the stem.

For example, construct a back-to-back stem-and-leaf plot for the data shown. The stem and leaf plot shows the number of goals scored by each team over the season.

Team 1: 2, 9, 35, 21, 20, 10, 6, 13, 24, 31, 3, 1 Team 2: 5, 6, 19, 12, 15, 22, 30, 28, 21, 14, 17, 8

Step 1: Put the numbers in each data set in order from smallest to largest

Writing the numbers in order we obtain:

Team 1: 1, 2, 3, 6, 9, 10, 13, 20, 21, 24, 31, 35 Team 2: 5, 6, 8, 12, 14, 15, 17, 19, 21, 22, 28, 30



Step 2: All apart from the final digit of each number is listed so that they appear in the central stem once

The digits listed in the stem will be the tens digits of each number. The ones digits of each number will go in the leaves.

The smallest value to list is a 1. It has zero tens and so, a 0 is written in the stem.

The largest values to list is a 35. It has 3 tens and so, the numbers in the stem will be listed from 0 up to 3.



Step 3: The final digits in data set 1 are listed to the left of the stem, getting larger as they get further from the stem

For example, entering the numbers with zero tens from team 1: we enter 1, 2, 3, 6 and 9.

These numbers are entered as 9 6 3 2 1 |0|.

The 0 in the stem shows that there is nothing in the tens column.

The numbers are listed backwards because they are listed so that the larger numbers are furthest from the stem.



The numbers of 10 and 13 are listed as 3 0 |1|.

Both numbers have a 1 in the tens column, so the stem has a 1 in it.

The numbers of 20, 21 and 24 are listed as 4 1 0 |2|.

The numbers of 31 and 35 are listed as 5 1 |3|.



Step 4: The final digits in data set 2 are listed to the right of the stem, getting larger as they get further from the stem

5, 6 and 8 are listed as |0| 5 6 8.

12, 14, 15, 17 and 19 are listed as |1| 2 4 5 7 9.

21, 22 and 28 are listed as |2| 1 2 8.

30 is listed as |3| 0.



Back-to-Back Stem-and-Leaf Plot with 3-Digit Numbers

When constructing a stem-and-leaf plot with 3-digit numbers, the hundreds and tens digits will go in the stem and the ones digit of each number will go in the leaves.

The back-to-back stem-and-lead plot shown below represents the heights (in centimetres) of two samples of plants.

Listing the values in sample 1:

101, 102, 102 and 108 are listed as 8 2 2 1 |10|.

This is because they all have a 1 and a 0 in their hundreds and tens columns respectively. Only the ones digit of each number is listed.

115 is listed as 5 |11|.

123, 125 and 129 are listed as 9 5 3 |12|.

134 and 137 are listed as 7 4 |13|.

140 and 145 are listed as 5 0 |14|.



Listing the values in sample 2:

103, 104, 108 and 108 are listed as |10| 3 4 8 8.

110 and 117 are listed as |11| 0 7.

123, 124, 124, 127 and 128 are listed as |12| 3 4 4 7 8.

132 is listed as |13| 2.

There are no plants in sample 2 with a 1 in the hundreds column and a 4 in the tens column.

Back-to-Back Stem-and-Leaf Plot with Decimals

Only the final decimal digit is listed in a stem-and-leaf plot. With decimal numbers, all other preceding digits are listed in the stem. The decimal point may also be listed in the stem.

For example, the following stem-and-leaf plot displays the heights of boys and girls in a class.

Listing the boy’s data:

1.49 is listed as |1.4| 9.

1.56 and 1.58 are listed as |1.5| 6 8.

1.60, 1.61, 1.64, 1.66 and 1.66 are listed as |1.6| 0 1 4 6 6.

1.82 is listed as |1.8| 2.



Listing the girl’s data:

1.39 is listed as 9 |1.3|.

1.46, 1.47 and 1.47 are listed as 7 7 6 |1.4|.

1.51 and 1.53 are listed as 3 1 |1.5|.

1.60, 1.62 and 1.63 are listed as 3 2 0 |1.6|.

How to Use a Back-to-Back Stem-and-Lead Plot to Compare Data Sets

Back-to-back stem-and-leaf plots are used to compare two data sets. They are a visual display of the shape of the data, so that skewness, range and modal values can more easily be seen. The sizes of the values within the two data sets can easily be compared by looking at their general position on the diagram.

For example, the back-to-back stem-and-leaf plot below shows the heights of boys and girls in a class. The mode, median, range and skewness of the data can easily be found from the diagram.



The mode from a stem-and-leaf plot is found by looking for the digit that is repeated the most in a particular row.

The mode for the girl’s data is seen to be 1.47 m. This is because the only digit that is repeated in this data set is a 7 in the |1.4| stem.

The mode for the boy’s data is 1.66 m. This is because the only digit that is repeated in this data set is a 6 in the |1.6| stem.

The mode is a type of average and since the boys have a larger mode, we can interpret this as the boys having a larger height on average.

The median is found in the middle of each data set on a stem-and-leaf plot.

There are 9 data values in each set of data. By crossing off the smallest 4 data values and the largest 4 data values, the middle value will be left. This is the median.

In the girl’s data, the values of 1.39, 1.46, 1.47 and 1.47 are crossed off at the bottom, along with 1.63, 1.62, 1.60 and 1.53 at the top.

This leaves 1.51 metres as the median height for the girls.

In the boy’s data, the values of 1.49, 1.56, 1.58 and 1.60 are crossed off at the bottom, along with 1.82, 1.66, 1.66 and 1.64 at the top.

This leaves 1.61 metres as the median height for boys.



The median is a type of average and so, we can say that the boys are taller on average.

The range is found on a stem-and-lead plot by subtracting the first value from the last value

The first value in the girl’s data is 1.39 m. The last value is 1.63 m.

The range of the girl’s data is 1.63 – 1.39 = 0.24 m.

The first value in the boy’s data is 1.49 m. The last value is 1.82 m.

The range of the boy’s data is 1.82 – 1.49 = 0.33.

The range describes the spread of the data. Since the boys have a larger range, the boy’s heights are more spread than the girls.

The skewness of the data on a stem-and-leaf plot can be found by drawing around the peaks of each row.

The skew of the data can be seen below.

The girl’s data forms two peaks and therefore we can call it bimodal.

The boy’s data starts with a slow climb and most of the data is bunched at the bottom. This is called negatively skewed data.

In the opposite case, where most of the data is bunched at the top, this would be called positively skewed data.



A stem-and-leaf plot easily shows the positions of the data values.

It is easy to see that the boy’s values are typically lower down in the diagram than the girl’s values.

This tells us that the boy’s values are typically found in the larger numbers on the diagram and so, it is easy to see that boys are taller than girls on average in this example.



Now try our lesson on Probability with Spinners where we learn how to understand probability with spinners.

Probability with Spinners

What is Solving by Inspection?

Solving an equation by inspection is a method used to obtain a solution without using algebra. The numbers in the equation are considered so that an estimate of the solution can be made. This guess is substituted into the equation to obtain an answer which is compared to the actual answer. Different values can be trialled until a solution is found.

For example, solve 𝑥 + 2 = 6 by inspection.

We think of the number that makes an answer of 6 when we add 2 to it.

We know that 4 is 2 less than 6 and so, 4 + 2 = 6.

Therefore 𝑥 must be equal to 4.

How to Solve an Equation by Inspection

To solve an equation by inspection, use the existing numbers in the equation to make a guess at the solution. Substitute this value into the equation and compare the result to the actual answer. If the answer is too small, a larger estimate should be trialled. If the answer is too large, a smaller answer should be trialled.

For example, solve 5𝑥-2=28.

We know that 𝑥 will be multiplied by 5 and then 2 will be subtracted to get an answer of 28.

Since 𝑥 is multiplied by 5, we look for a number that when multiplied by 5 gives us a result close to 28.

We know that 5 × 6 = 30.

5 × 6 – 2 = 28 and so, 𝑥 = 6.

Here is another example.

Solve 2𝑥 + 3 = 19 by inspection.

This time we will substitute values into the equation with the goal of finding an answer of 19.

Since we are multiplying 𝑥 by 2, we can think of a multiple of 2 that is close to the answer of 19.

2 × 10 = 20 and so, we can try this.

2 × 10 + 3= 23.

This answer is too large and so, we try a smaller value of 𝑥.

We can substitute 𝑥 = 7 so that 2 × 7 + 3 = 17.

This is too small so we try a larger value of 𝑥.

We try 𝑥 = 8, which gives us 2 × 8 + 3 = 19.

This gives us the correct answer and so, 𝑥 = 8.

What are Near Doubles?

Near doubles are maths facts that are close to doubles maths facts. For example, 3+4 is a near double because it is one more than 3+3. Near doubles are typically one or two larger or smaller than doubles facts.

Since we know that 3 + 3 = 6, we can work out 3 + 4 since it will be one larger.

3 + 4 = 7.

How to do Near Doubles Addition

To add numbers using the near doubles strategy, double one of the numbers in the addition. Then add or subtract the difference between the other number and the number that was doubled.

For example, add 6 + 7.

Doubling 6, we get 12.

Looking at the other number, 7 is one larger than 6 and so, we must add one more to our answer.

6 + 7 = 13.

5 + 7 is a near double as it is close to 5 + 5.

Using the known double fact that 5 + 5 = 10, 5 + 7 must be two more than this and so, 5 + 7 = 12.

Near doubles addition may be used when the numbers are below the doubles facts.

For example, 8 + 7 is close to the doubles fact 8 + 8 = 16. However, since 7 is one less than 8, we subtract one from the answer.

Therefore 8 + 7 = 15.

Teaching the Near Doubles Strategy

When teaching near doubles, first teach the doubles facts. Knowing these is necessary for the near doubles method to be effective. Using tens frames can be an effective teaching strategy for showing the comparative size of near doubles.

The near doubles addition strategy uses the knowledge of doubles facts as a crutch. This is because doubles facts are generally easily memorised.

The doubles facts that should be learnt are:

• 1 + 1 = 2
• 2 + 2 = 4
• 3 + 3 = 6
• 4 + 4 = 8
• 5 + 5 = 10
• 6 + 6 = 12
• 7 + 7 = 14
• 8 + 8 = 16
• 9 + 9 = 18
• 10 + 10 = 20

Tens frames are useful for teaching the concept of a ‘near double’.

In the tens frame below, 3 + 3 is represented with pink counters.

It can be seen that 3 + 4 is shown with an extra green counter. The green counter is used to represent the difference needed.

This can help conceptualise the idea of finding a difference to a known doubles fact.

It would be beneficial to first put the pink counters in place to represent the double. Then calculate the value of the double before adding the extra counters and then counting on from the previous total.

For example, 4 + 4 = 8 can be shown.

An extra counter can be placed next to one of the fours to make 4 + 5 = 9.

The idea is not to count the whole total again but simply count on from the known doubles fact of 4 + 4 = 8.

Once the conceptual idea of a near double has been taught, it is beneficial to practise them using tasks that require recall such as flashcards.

Near Doubles Strategy for Subtraction

The near doubles strategy can be used for subtraction when the number being subtracted is close to half of the larger number. Use a known doubles fact and correct the result afterwards. For example, 12 – 6 = 6 and so 12 – 7 requires a further one to be subtracted. Therefore 12 – 7 = 5.

For example, 15 – 8 can be calculated using the near doubles subtraction strategy.

We know that 16 – 8 = 8 because 8 doubled is 16.

15 – 8 must be one smaller than 16 – 8.

Therefore 15 – 8 = 7.

What is the Partial Sums Method?

The partial sums method is an addition strategy in which the numbers are added together in parts according to their place value. For example, 635 + 312 is calculated by adding 600 + 300 = 900, 30 + 10 = 40 and 5 + 2 = 7. These partial sums are then added as 900 + 40 + 7 = 947.

• Adding the hundreds

600 + 300 = 900

• Adding the tens

30 + 10 = 40

• Adding the ones

5 + 2 = 7

• Adding the partial sums

900 + 40 + 7 = 947

Partial Sums Method Examples

Here are some examples of using the partial sums method.

Addition Sum	Sum of the Hundreds	Sum of the Tens	Sum of the Ones	Adding the Partial Sums
346 + 523	300 + 500 = 800	40 + 20 = 60	6 + 3 = 9	800 + 60 + 9 = 869
105 + 382	100 + 300 = 400	0 + 80 = 80	5 + 2 = 7	400 + 80 + 7 = 487
645 + 115	600 + 100 = 700	40 + 10 = 50	5 + 5 = 10	700 + 50 + 10 = 760
361 + 155	300 + 100 = 400	60 + 50 = 110	1 + 5 = 6	400 + 110 + 6 = 516
269 + 363	200 + 300 = 500	60 + 60 = 120	9 + 3 = 12	500 + 120 + 12 = 632

Subtraction with Partial Differences

The partial differences subtraction method involves subtracting each place value column individually and then finding the total of these results. Always subtract the smaller number from the larger number. If the first number is subtracted from the second number, the result will be negative.

Use the partial differences method to subtract 625 – 453.

625 is made up of 600 + 20 + 5.

453 is made up of 400 + 50 + 3.

• Subtracting the hundreds

600 – 400 = 200

• Subtracting the tens

20 – 50 = -30.

Alternatively, 50 – 20 = 30 but since the numbers were subtracted the other way around, we write -30.

• Subtracting the ones

5 – 3 = 2

• Finding the total

200 – 30 + 2 = 172.

Therefore 625 – 453 = 172.

Here is another example of using the partial differences method of subtraction.

Calculate 814 – 366.

• Subtracting the hundreds

800 – 300 = 500

• Subtracting the tens

We need to subtract the numbers in the opposite order and therefore, the result will be negative.

60 – 10 = -50.

• Subtracting the ones

Again, we need to subtract the numbers in the opposite order and therefore, the result will be negative.

6 – 4 = -2.

• Finding the total

500 – 50 – 2 = 448.

Therefore 814 – 366 = 448.

How to Find the Difference Between Two Numbers

To find the difference between two numbers, subtract the smaller number from the larger number. For example, the difference between 5 and 3 is 2 because 5 – 3 = 2.

No matter how large the numbers are, subtracting the smaller number from the larger number will always calculate the difference.

For example, find the difference between 80 and 20.

The larger number is 80 and the smaller number is 20.

Therefore to calculate the difference between the numbers, subtract 20 from 80.

80 – 20 = 60 and so, the difference between 80 and 20 is 60.

Teaching How to Find the Difference Between Two Numbers

To teach finding the difference:

1. Use physical objects such as counters to help count on
2. Use number lines to show the size of numbers
3. Show that subtraction gives us the difference

When introducing the idea of a difference between numbers, it is best to start with very small numbers that can be represented with physical objects, such as counters.

Placing the counters side by side can be used to show the difference in size between one number and another.

For example, in the image below we can see that 4 contains 3 more counters than 1 and so, the difference between 4 and 1 must be 3.

After the idea of what the difference between two numbers means is understood, it is time to increase the size of the numbers and look at numbers on a number line.

Number lines are useful tools to use for teaching the difference between two numbers. Mark both numbers on the number line and count the number of jumps needed to move from the smaller number to the larger number.

For example, the difference between 6 and 3 can be seen to be 3 because there are 3 jumps needed to move from the smaller number to the larger number.

How to Find the Difference Between Two Numbers Formula

The formula for the difference between two numbers, a and b is a-b. Where a is the largest number and b is the smallest number.

For example, the difference between 25 and 13.

The Difference Between Two-Digit Numbers

To find the difference between two-digit numbers, subtract the smaller number from the larger. With two-digit numbers, the subtraction is most easily calculated using vertical column subtraction.

For example, calculate the difference between 56 and 33.

Line up the digits and subtract vertically.

In the ones column, 6 – 3 = 3.

In the tens column, 5 – 3 = 2.

56 – 33 = 23.

How to Find the Difference Between Two Odd Numbers

The difference between two odd numbers is always even. For example, 25 – 13 – 12, which is even.

All odd numbers have a difference which is even. This is because consecutive odd numbers always differ by 2. Adding 2 to an odd number takes us to the next odd number.

How to Find the Difference Between Two Even Numbers

The difference between two even numbers is always even. For example, 16 – 12 = 4, which is even.

All even numbers have a difference which is even. This is because even numbers are all multiples of 2. Every even number is in the two times table. To get from one even number to the next, add two.

How to Find the Difference Between Two Negative Numbers

The difference between two negative numbers is the same as the difference between these numbers with positive signs. For example the difference between -8 and -1 is 7 because the difference between 8 and 1 is 7.

Why Estimating is Important

In maths, estimation means to simplify numbers in a calculation in order to get a close answer more quickly and easily than doing the full calculation. The benefits of estimation are that it can often be completed mentally and the result can be used to check the result of a calculation.

For example, if you need to buy 5 pens for $2.99 each, it is easier to find 5 × $3 than work out 5 × $2.99.

The correct cost is $14.95 which is very close to the estimation of $15.

This estimation can be completed quickly and easily in your head.

How to Estimate to the Nearest Integer

Estimating a number to the nearest integer means to find the nearest whole number to it. Look at the digit in the tenths place. If it is 5 or more, round up. If it is 4 or less round down. For example, 6.27 rounds down to 6 because there is a 2 in the tenths place.

An integer is a whole number.

When rounding to the nearest integer, only look at the digit immediately after the decimal point in the tenths column to decide whether to round up or down.

For example, estimate 2.83 to the nearest integer.

The digit in the tenths column is 8. This is ‘5 or more’ and so we round up.

We round 2.87 up to 3 which means that we estimate 2.87 to be 3.

How to Estimate to the Nearest Ten

To estimate a number to the nearest ten, look at the digit in the ones column. If it is 5 or more, round up. If it is 4 or less, round down. For example, 14 rounds down to 10 because there is a 4 in the ones column.

For example, estimate 55 to the nearest ten.

There is a 5 in the ones column and so, we round up.

How to Estimate to the Nearest Hundred

To estimate a number to the nearest hundred, look at the digit in the tens column. If it is 5 or more, round up. If it is 4 or less, round down. For example, 247 rounds down to 200 because there is a 4 in the tens column.

For example, estimating 1363 to the nearest hundred is 1400 because 6 in the tens column is ‘5 or more’.

How to Estimate an Answer

When estimating an answer use the following rules:

• Focus on the digits at the start of each number as they have a greater impact on the answer.
• Round each number greater than one to the nearest whole number, ten, hundred or thousand.
• Round any number less than one to the nearest fractional amount.
• Round all numbers before performing the calculation.

For example, estimate the size of the answer to 39 × 4.85.

39 rounds up to 40. Multiples of 10 are generally easier to multiply.

4.85 rounds up to 5. Five is chosen as it is also an easier number to multiply.

Since 4 × 5 = 20, 40 × 5 = 200. We need to add another zero.

Here is an example of estimating the total cost of a shopping list.

When estimating money, round each amount to the nearest whole number.

• $0.90 rounds up to $1.
• $1.25 rounds down to $1.
• $2.87 rounds up to $3.
• $6.10 rounds down to $6.
• $3.22 rounds down to $3.

Adding up the total we have $1 + $1 + $3 + $6 +$3 = $14.

How to Estimate an Addition

To estimate an addition, round all numbers to their first digit before adding them. To do this, look at the second digit of each number. If this digit is 5 or more, round up. If it is 4 or less, round down.

For example, estimate the addition of 48 + 51.

The second digit of 48 is 8, therefore we round it up to 50.

The second digit of 51 is 1, therefore we round it down to 50.

Perform the addition after the rounding. 50 + 50 = 100, therefore an estimate to 48 + 51 is 100.

The correct answer is 99, which is only 1 off 100.

Here is another example of estimating an addition.

Estimate 384 + 209.

The second digit of 384 is 8, therefore it rounds up to 400.

The second digit of 209 is 0, therefore it rounds down to 200.

400 + 200 = 600, therefore the estimate to the addition of 384 + 209 = 600.

The correct answer to 384 + 209 is 593, which is only 7 away from the estimate of 600.

How to Estimate a Subtraction

To estimate a subtraction, round each number to its first digit and then subtract. To do this, look at the second digit of each number. If it is 5 or more, round up. If it is 4 or less, round down.

For example, estimate the subtraction of 73 – 29.

The second digit of 73 is a 3, therefore 73 rounds down to 70.

The second digit of 29 is 9, therefore 29 rounds up to 30.

The estimate of 73 – 29 is 70 – 30, which equals 40.

The correct answer to 73 – 29 is 44, which is 4 away from the estimated answer of 40.

Subtraction is used to find a difference.

Find the estimated difference between 988 and 674.

The second digit of 988 is 8, therefore 988 rounds up to 1000.

The second digit of 674 is 7, therefore 674 rounds up to 700.

1000 – 700 = 300, therefore the estimated difference between 988 and 674 is 300.

The difference between 988 and 674 is 314. This is 14 away from the estimated difference of 300.

How to Estimate Multiplication

To estimate the answer to a multiplication, round the numbers to their first digit or to easy to multiply digits before multiplying them. For example, 39 × 4.85 can be estimated as 40 × 5 which equals 200. The correct answer is 189.15.

Here is another example of estimating multiplication.

Estimate 482 × 734.

482 can be estimated as 500.

734 can be estimated as 700.

To calculate 500 × 700, multiply 5 × 7 to get 35 and add on the four zeros found in 500 and 700.

500 × 700 = 350000 and so, the estimate to 482 × 734 is 350000.

The correct answer is 353788.

How to Estimate Division

To estimate division, find similar numbers that can be divided exactly. For example 15 ÷ 4 can be estimated as 16 ÷ 4 = 4. 15 is close to 16 and 16 is chosen because it can be divided exactly by 4.

For example, estimate the division 194592 ÷ 4126.

In this example, 194592 rounds to 200000 and 4126 can be rounded to 4000.

We can cancel the three zeros in 4000 with the three zeros in 200000 to leave 200 ÷ 4.

20 ÷ 4 = 5 and so 200 ÷ 4 = 50.

Here is another example of estimating division.

Estimate 19 ÷ 3.

Instead of rounding 19 up to 20, it is best to round it down to 18. We choose the nearest number that can be divided exactly by 3.

18 ÷ 3 = 6 and so, 19 ÷ 3 is just a little larger than 6.

19 ÷ 3 = 6.33.

How to Estimate Decimals

When calculating with decimals, try to round the decimal number to the nearest whole number. For decimals less than one whole, round the decimal to a number that is equivalent to a simple fraction.

Here are some useful decimals and their fraction equivalents.

Decimal	Equivalent Fraction
0.1	1/10
0.2	1/5
0.25	1/4
0.33	1/3
0.4	2/5
0.5	1/2
0.6	3/5
0.66	2/3
0.75	3/4
0.8	4/5

For example, estimate 0.26 × 23.87.

0.26 can be estimated as 0.25, which is the same as ¹/₄.

28.87 can be estimated as 24.

0.26 × 23.87 can be estimated as ¹/₄ of 24 which equals 6.

How to Estimate Decimals When Adding

To estimate addition involving decimals, round each decimal to the nearest whole number and then add them.

For example, estimate 1.85 + 14.03 + 3.92.

2 + 14 + 4 = 20.

How to Estimate Decimal Subtraction

To estimate subtraction involving decimals, round each decimal to the nearest whole number and then subtract them.

For example, estimate 14.99 – 2.85.

Rounding to the nearest whole number this can be estimated as 15 – 3 = 12.

How to Estimate Decimal Multiplication

To estimate a decimal multiplication:

• Round any number greater than one to the nearest whole number.
• Round any decimal less than one to a decimal that has a fraction equivalent.

For example estimate 0.32 × 59.3.

0.32 can be estimated as 0.33, which is equivalent to ¹/₃.

59.3 can be estimated as 60.

0.32 × 59.3 can be estimated to be ¹/₃ of 60, which equals 20.

The exact answer is 18.976.

How to Estimate Decimal Division

To estimate a decimal division, round the numbers so that the division can be done exactly. For example, estimate 7.9 ÷ 2.03. The numbers can be estimated as 8 ÷ 2 and so the division can be estimated to be equal to 4

The exact answer is 3.89.

How to Estimate to the Nearest Tenth

To estimate a number to the nearest tenth, look at the digit in the hundredths column. If it is 5 or more, add 1 to the digit in the tenths column. If it is 4 or less, keep the tenths digit the same. Remove all digits that follow.

For example estimate 0.5814 + 2.632.

There is an 8 in the hundredths column of 0.5814 and so, 0.5814 rounds up to 0.6.

There is a 3 in the hundredths column of 2.632 and so, 0.2632 rounds down to 2.6.

0.6 + 2.6 = 3.2 and so the estimate for this calculation is 3.2.

The correct answer is 3.2134.

How to Estimate Percentages

To estimate a percentage:

• Estimate what 50% is by dividing the total by 2.
• Estimate what 10% is by dividing the total by 10.
• Estimate what 5% is by finding half of 10%.
• Estimate what 1% is by dividing the total by 100.
• Use combinations of these percentages to find a similar percentage to that required.

For example, estimate 52% of 40.

52% is very similar to 50%. Therefore an estimate of 52% of 40 will be approximately half of 40, which is 20.

We know that 52% will be a little larger than 50% and so, we can estimate it as larger than 20.

1% is found by dividing 40 by 100 to get 0.4 and so, doubling this 2% is 0.8.

Therefore 52% is 20.8.

For example, estimate 16% of $20.

16% is similar to 15% and so, can be estimated by adding 10% and 5%.

10% is found by dividing $20 by 10 to get $2 and then 5% is half of 10%, which is $1.

15% is therefore $2 + $1 = $3. Therefore 16% is a little larger than $3.

To find 16%, add 1% to the 15% found previously.

1% is found by dividing $20 by 100 to get $0.20.

Therefore 16% is $3.20.

How to Estimate with Fractions

To estimate with fractions, round each fraction to the nearest whole number. If the fraction is less than one whole, compare the numerator to the denominator so that the fraction can be compared to a fraction that is easier to work with.

For example, estimate ¹⁴/₂₉ of 14.

14 is exactly half of 28 and so, ¹⁴/₂₉ can be estimated as ¹/₂.

¹/₂ of 14 is 7.

How to Estimate Fractions When Adding

When estimating adding fractions, compare the numerator to the denominator. If the numerator is close to the denominator, estimate the fraction as one whole. If the numerator is much less than the denominator, estimate the fraction as zero. If the numerator is similar to half of the denominator, estimate the fraction as one half.

For example, estimate ⁵/₆ + ¹⁷/₂₀ + ⁵/₉ + ¹/₁₁.

⁵/₆ can be estimated as 1 whole because 5 is close to 6.

¹⁷/₂₀ can also be estimated as 1 whole because 17 is close to 20.

⁵/₉ can be estimated as one half because 5 is half of 10, which is close to half of 9.

¹/₁₁ can be estimated as zero since 1 is much less than 11.

Therefore ⁵/₆ + ¹⁷/₂₀ + ⁵/₉ + ¹/₁₁ can be estimated as 1 + 1 + 0.5 + 0, which equals 2.5

The correct answer is 2.33, which is close to the estimate of 2.5.

How to Estimate Fractions When Subtracting

To estimate fractions when subtracting, compare the numerator to the denominator. If the numerator is close to the denominator, estimate the fraction as one whole. If the numerator is much less than the denominator, estimate the fraction as zero. If the numerator is similar to half of the denominator, estimate the fraction as one half.

For example, estimate 3⁷/₈ – ⁶/₁₁.

⁷/₈ can be estimated as one whole since 7 is close to 8. Therefore 3⁷/₈ can be estimated as 4.

⁶/₁₁ can be estimated as ¹/₂ since 6 is close to half of 11.

Therefore 3⁷/₈ – ⁶/₁₁ can be estimated as 4 – 0.5 which equals 3.5.

The correct answer is approximately 3.33 which is close to the estimate of 3.5.

How to Estimate Fractions When Multiplying

To estimate multiplication with fractions, compare each fraction to one half. If the fraction is greater than or equal to one half, round it up to the next whole number. If the fraction is less than one half, round it down to the previous whole number. Then multiply the whole numbers together.

For example, estimate 4⁴/₅ × 2¹/₄.

4⁴/₅ can be estimated as 5 since ⁴/₅ is larger than ¹/₂.

2¹/₄ can be estimated as 2 since ¹/₄ is less than ¹/₂.

4⁴/₅ × 2¹/₄ can be estimated as 5 × 2 = 10.

The exact answer is 10.8 which is close to the estimate of 10.

How to Estimate Fractions When Dividing

To estimate division with fractions, compare each fraction to one half. If the fraction is greater than or equal to one half, round it up to the next whole number. If the fraction is less than one half, round it down to the previous whole number. Then divide the whole numbers.

For example, estimate 3³/₄ ÷ 2¹/₅.

3³/₄ can be estimated as 4 since ³/₄ is larger than one half.

2¹/₅ can be estimated as 2 since ¹/₅ is less than one half.

3³/₄ ÷ 2¹/₅ can be estimated as 4 ÷ 2 which equals 2.

The correct answer is approximately 1.70 which is close to the estimate of 2.