# 8.4: Estimation by Rounding Fractions

Learning Objectives

• be able to estimate the sum of two or more fractions using the technique of rounding fractions

Estimation by rounding fractions is a useful technique for estimating the result of a computation involving fractions. Fractions are commonly rounded to (dfrac{1}{4}), (dfrac{1}{2}), (dfrac{3}{4}), 0, and 1. Remember that rounding may cause estimates to vary.

Sample Set A

Make each estimate remembering that results may vary.

Estimate (dfrac{3}{5} + dfrac{5}{12}).

Solution

Notice that (dfrac{3}{5}) is about (dfrac{1}{2}), and that (dfrac{5}{12}) is about (dfrac{1}{2}).

Thus, (dfrac{3}{5} + dfrac{5}{12}) is about (dfrac{1}{2} + dfrac{1}{2} = 1). In fact, (dfrac{3}{5} + dfrac{5}{12} = dfrac{61}{60}), a little more than 1.

Sample Set A

Estimate (5 dfrac{3}{8} + 4 dfrac{9}{10} + 11 dfrac{1}{5}).

Solution

Adding the whole number parts, we get 20. Notice that (dfrac{3}{8}) is close to (dfrac{1}{4}), (dfrac{9}{10}) is close to 1, and (dfrac{1}{5}) is close to (dfrac{1}{4}). Then (dfrac{3}{8} + dfrac{9}{10} + dfrac{1}{5}) is close to (dfrac{1}{4} + 1 + dfrac{1}{4} = 1 dfrac{1}{2}).

Thus, (5 dfrac{3}{8} + 4 dfrac{9}{10} + 11 dfrac{1}{5}) is close to (20 + 1 dfrac{1}{2} = 21 dfrac{1}{2}).

In fact, (5 dfrac{3}{8} + 4 dfrac{9}{10} + 11 dfrac{1}{5} = 21 dfrac{19}{40}), a little less than (21 dfrac{1}{2}).

Practice Set A

Use the method of rounding fractions to estimate the result of each computation. Results may vary.

(dfrac{5}{8} + dfrac{5}{12})

Results may vary. (dfrac{1}{2} + dfrac{1}{2} = 1). In fact, (dfrac{5}{8} + dfrac{5}{12} = dfrac{25}{24} = 1 dfrac{1}{24})

Practice Set A

(dfrac{7}{9} + dfrac{3}{5})

Results may vary. (1 + dfrac{1}{2} = 1 dfrac{1}{2}). In fact, (dfrac{7}{9} + dfrac{3}{5} = 1 dfrac{17}{45})

Practice Set A

(8 dfrac{4}{15} + 3 dfrac{7}{10})

Results may vary. (8 dfrac{1}{4} + 3 dfrac{3}{4} = 11 + 1 = 12). In fact, (8 dfrac{4}{15} + 3 dfrac{7}{10} = 11 dfrac{29}{30})

Practice Set A

(16 dfrac{1}{20} + 4 dfrac{7}{8})

Results may vary. ((16 + 0) + (4 + 1) = 16 + 5 = 21). In fact, (16 dfrac{1}{20} + 4 dfrac{7}{8} = 20 dfrac{37}{40})

## Exercises

Estimate each sum or difference using the method of rounding. After you have made an estimate, find the exact value of the sum or difference and compare this result to the estimated value. Result may vary.

Exercise (PageIndex{1})

(dfrac{5}{6} + dfrac{7}{8})

(1 + 1 = 2(1 dfrac{17}{24}))

Exercise (PageIndex{2})

(dfrac{3}{8} + dfrac{11}{12})

Exercise (PageIndex{3})

(dfrac{9}{10} + dfrac{3}{5})

(1 + dfrac{1}{2} = 1 dfrac{1}{2} (1 dfrac{1}{2}))

Exercise (PageIndex{4})

(dfrac{13}{15} + dfrac{1}{20})

Exercise (PageIndex{5})

(dfrac{3}{20} + dfrac{6}{25})

(dfrac{1}{4} + dfrac{1}{4} = dfrac{1}{2} (dfrac{39}{100}))

Exercise (PageIndex{6})

(dfrac{1}{12} + dfrac{4}{5})

Exercise (PageIndex{7})

(dfrac{15}{16} + dfrac{1}{12})

(1 + 0 = 1 (1 dfrac{1}{48}))

Exercise (PageIndex{8})

(dfrac{29}{30} + dfrac{11}{20})

Exercise (PageIndex{9})

(dfrac{5}{12} + 6 dfrac{4}{11})

(dfrac{1}{2} + 6 dfrac{1}{2} = 7 (6 dfrac{103}{132}))

Exercise (PageIndex{10})

(dfrac{3}{7} + 8 dfrac{4}{15})

Exercise (PageIndex{11})

(dfrac{9}{10} + 2 dfrac{3}{8})

(1 + 2 dfrac{1}{2} = 3 dfrac{1}{2} (3 dfrac{11}{40}))

Exercise (PageIndex{12})

(dfrac{19}{20} + 15 dfrac{5}{9})

Exercise (PageIndex{13})

(8 dfrac{3}{5} + 4 dfrac{1}{20})

(8 dfrac{1}{2} + 4 = 12 dfrac{1}{2} (12 dfrac{13}{20}))

Exercise (PageIndex{14})

(5 dfrac{3}{20} + 2 dfrac{8}{15})

Exercise (PageIndex{15})

(9 dfrac{1}{15} + 6 dfrac{4}{5})

(9 + 7 = 16 (15 dfrac{13}{15}))

Exercise (PageIndex{16})

(7 dfrac{5}{12} + 10 dfrac{1}{16})

Exercise (PageIndex{17})

(3 dfrac{11}{20} + 2 dfrac{13}{25} + 1 dfrac{7}{8})

(3 dfrac{1}{2} + 2 dfrac{1}{2} + 2 = 8) (7 (dfrac{189}{200}))

Exercise (PageIndex{18})

(6 dfrac{1}{12} + 1 dfrac{1}{10} + 5 dfrac{5}{6})

Exercise (PageIndex{19})

(dfrac{15}{16} - dfrac{7}{8})

(1 - 1 = 0 (dfrac{1}{16}))

Exercise (PageIndex{20})

(dfrac{12}{25} - dfrac{9}{20})

#### Exercises for Review

Exercise (PageIndex{21})

The fact that

(( ext{a first number } cdot ext{a second number}) cdot ext{a third number} = ext{a first number } cdot ( ext{a second number } cdot ext{a third number}))

is an example of which property of multiplication?

associative

Exercise (PageIndex{22})

Find the quotient: (dfrac{14}{15} div dfrac{4}{45}).

Exercise (PageIndex{23})

Find the difference: (3 dfrac{5}{9} - 2 dfrac{2}{3})

(dfrac{8}{9})

Exercise (PageIndex{24})

Find the quotient: (4.6 div 0.11).

Exercise (PageIndex{25})

Use the distributive property to compute the product: (25 cdot 37).

(25(40 - 3) = 1000 - 75 = 925)

## Fractions Estimating (Estimating Sums and Differences of Fractions Calculator)

Make use of this Free online & handy Factions Estimating Tool & directly find the estimated sum or difference of fractions result in split seconds. Simply enter the positive proper fractions in the input field and then click on the calculate button.

Here are some samples of Fractions Estimating calculations.

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## 8.4: Estimation by Rounding Fractions

Online lessons on rounding and estimating.

In-depth notes and examples on rounding and estimating

This site links you to other references on rounding.

This site links you to other references on estimating, rounding, and money.

This is an awesome site from Scholastic that will really help you with rounding whole numbers.

##### Rounding (Games)

Select the number that is rounding to the correct place

Round the numbers to the given place

Round each number to the nearest thousand then add or subtract

Test your memory to match the cards to the closest rounded value.

Test your skills and see if you can select the correct answer.

Answer the questions correctly to score a basket. Try to score as many points possible in 90 seconds.

##### Estimating (Games)

Estimate with addition to take as many monkeys up the elevator as possible in three minutes.

Estimate the problems to score a home run.

Estimate the problems accurately to win the game of golf.

Try to connect four pieces in a row first. However, you must answer questions using estimation to.

## Estimating Fractions

An exclusive page that contains plentiful worksheets on rounding proper fraction, improper fraction and mixed number to the nearest whole number.

Each estimating fraction worksheet pdf has practice problems on rounding mixed number to the nearest whole number and to thereby find the sum or difference.

Round the mixed numbers to the nearest whole and estimate the product. Compare the product using greater than or less than symbol.

Estimate the product by rounding each mixed number in these printable worksheets to the nearest whole number.

In these estimating quotient worksheets, round each mixed number to the nearest whole number and then divide to estimate the quotient. The dividend is always greater than the divisor.

## Estimating in Multiplication

This is a complete lesson with teaching and exercises about estimation in multiplication, meant for fourth grade. First, students practice rounding two- and three-digit numbers and money amounts and estimating products (answers to multiplication problems). Then they solve many word problems that involve estimating.

If you don't need an exact result, you can estimate. To estimate a multiplication, round some or all of the factors so that it will be easy to multiply mentally.

189 can be rounded to 200.

The estimated product is 8 × 200 = 1,600.

The estimated product is 40 × 80 = 3,200.

Round the numbers to 20 and $4.50. Multiply in parts: 20 ×$4 = $80 and 20 × 50¢ = 1000¢ =$10. Then add: $80 +$10 = $90. 1. Estimate by rounding one or both factors. Don't round both if you can calculate in your head just by rounding one factor! 2. Estimate the cost. Round one or both numbers so you can multiply in your head! a. 24 chairs at$44.95 per chair

b. 512 Popsicles at 19¢ each

d. Six tennis balls that cost $3.37 each and two rackets that cost$11.90 each.

Example. If each bus can seat 57 passengers, how many buses do you need to seat 450 people?

 One bus seats 57 passengers. Two buses seat 114 passengers. Ten buses seat 570 passengers. Eight buses seat 8 × 57 passengers.

With how many buses will your answer be 450 or a little more?

This problem could be solved by division (450 ÷ 57) but instead, you can estimate using multiplication. Round the number 57 to 60, and quickly calculate:

7 × 60 = 420 and 8 × 60 = 480. It looks like 8 buses are needed for 450 people.

3. Solve the problems using estimation.

a. An advertisement in a newspaper costs $349. How many ads can Bill buy with$2000?

b. Renting skates at a skating rink costs $2.85 per hour. How many whole hours can Sandra skate for$25?

c. A can of beans costs .29. A bag of lentils costs .42.
Estimate which is cheaper: to buy eight cans of beans
or to buy five bags of lentils.

d. Jackie needs to buy 8 ft of string for each of
the 28 students in the craft class.
The string costs .22 per foot. Estimate her total cost.

This lesson is taken from Maria Miller's book Math Mammoth Multiplication 2, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.

#### Math Mammoth Multiplication 2

A self-teaching worktext for 4th grade that covers multiplying by whole tens and hundreds, multi-digit multiplication in columns, order of operations, word problems, scales problems, and money problems.

## When to Use Estimates?

Imagine you organize a carnival in your town.

The first thing you do is find out roughly how many guests will visit the carnival.

Can you calculate the exact number of visitors attending the carnival? It is practically impossible.

Let us consider another situation.

The finance minister of a certain country presents an annual budget.

The minister allocates a certain amount under the head "Education."

Can the amount allocated be absolutely accurate?

It can only be a reasonably good estimate of the expenditure the country needs for education during that year.

In the above-mentioned scenarios, a rough estimate will help us plan things properly.

Here we do not need exact figures.

In all these kinds of situations where the round-off value is as good as the exact value, we use estimates.

## Estimating Differences

A difference is the answer to a subtraction problem. It is best to estimate before subtracting decimals and fractions. You can use an estimate to check the accuracy of the difference when adding decimals or fractions.

### Find the difference of

1. Round each decimal to the nearest whole number.
2. 95.16 is rounded to 95 and 16.73 is rounded to 17.
3. Subtract: 95-17 = 78 , so the actual difference should be close to 78.
4. Use this to check the actual difference.
5. 95.16 – 16.73 = 78.43

The difference 78.43 is very close to the estimate 78, so the answer is reasonable and probably correct.

## 7.1 Estimation

Estimation is the process of making a guess about the size or cost of something, without doing the actual measurement or calculation. Estimations are sometimes needed because we do not have all the facts, or because we are pressed for time. Estimation can also help you to see whether the answer to a problem makes sense.

estimation Estimation is the process of making a guess without doing the actual measurement or calculation.

### Estimating distances and dimensions

A distance can be described as the length of the space between two points. An example of a distance is the length of the path between the door of the classroom and the door to the principal's office.

The dimensions of an object are the measurable lengths that we use to determine the size of the object. Examples of dimensions are the length, breadth (or width) and height of a box.

distance Distance is the length of the space between two points.

dimensions The dimensions of an object are the measurable lengths that we use to determine the size of the object, such as its length, breadth and height.

When we estimate distances and dimensions, it is important to use the correct unit of measurement. The SI unit for length is the metre. A length of one metre (1 m) is more or less equal to:

• the height of a 6-year-old child
• a large step of an adult
• the length of a prayer rug.

Other units of measurement used often are multiples of ten of 1 m. The most common multiples are shown in the table below.

SI unit An SI unit is a basic unit of measurement that forms part of the International System of Units.

### Worked example 7.1: Evaluating an estimation of a dimension

Adebankole is 14 years old. He estimated his own height as 160 cm. Evaluate his estimation.

Step 1: Evaluate whether the unit of measurement is appropriate.

Look at your ruler to see how long 1 cm is. It is quite short compared to the height of a teenager.

It might be better to use metres, but then Adebankole would need to use a decimal point in his estimation. To avoid the decimal point, centimetres is an appropriate unit.

Step 2: Evaluate whether the value of the estimation makes sense.

The height of a 6-year-old child is roughly 1 m. That is equal to 100 cm. So Adebankole's height is most probably more than 100 cm.

Most humans are not taller than 2 m, which is equal to 200 cm.

Adebankole's estimation is between 100 cm and 200 cm, so it makes sense.

### Exercise 7.1: Estimate distances and dimensions in your classroom

Estimate the following distances and dimensions. Include an appropriate unit of measurement in each case. Then use a ruler, metre rule or tape measure to get the real measurements. Compare the real measurements to your estimations.

3. Length of your Mathematics workbook
4. Height of your thickest textbook
5. Length of the classroom
6. Width of the classroom
7. Width of the blackboard
8. Height of your teacher's table
9. Width of the classroom door
10. Distance from the doorknob of the Mathematics classroom to the doorknob of the classroom next to it

### Exercise 7.2: Estimate everyday distances and dimensions

Estimate the following distances and dimensions by deciding which one of the options a) to d) is the most correct.

Height of a standard door

Maximum length of a standard football field

Shortest driving distance from Murtala Muhammed International Airport to Ibadan

### Estimate capacity

Volume is the amount of space occupied by an object or the amount of space inside a container. The amount of liquid that a container can hold is called the capacity of the container.

capacity The amount of liquid that a container can hold is called the capacity of the container.

The SI unit for capacity is the litre. A capacity of one litre (1 L) is more or less equal to:

Other units of measurement used often are multiples of ten of 1 L. The most common multiples are shown in the table below.

### Worked example 7.2: Evaluating an estimation of capacity

Amaka is feeling sick. The doctor gives her a bottle of medicine. She estimates that the capacity of the bottle is 250 cl. Evaluate her estimation.

Step 1: Evaluate whether the unit of measurement is appropriate.

Think about how much liquid 1 cl is. It is about two teaspoons.

One teaspoon is about 5 ml. The instructions on a bottle of medicine normally tells you how many teaspoons to take at time. Therefore, both millilitres and centilitres are appropriate units.

Step 2: Evaluate whether the value of the estimation makes sense.

If there is 250 cl in the bottle, it means the bottle holds 500 teaspoons of medicine.

At two teaspoons twice a day, it would take days to finish the medicine. That is more than 4 months! It is not possible to get so much medicine from a normal medicine bottle.

Amaka's estimation does not make sense. An estimation of 25 cl, which is 250 ml, would be a better estimation.

### Exercise 7.3: Estimate capacity of containers

You might use the containers listed below at school or at home. Estimate the capacity of the containers. Include an appropriate unit of measurement in each case. Then use measuring cylinders from the science laboratory, or a set of measuring spoons and cups used for baking, to get the real measurements. Compare the real measurements to your estimations.

### Exercise 7.4: Estimate capacity of everyday containers

Estimate the capacity of the following containers by deciding which one of the options a) to d) is the most correct.

Petrol tank of a small car

Petrol tanker (large truck that transports petrol)

### Estimate mass

The mass of an object is a measure of how much matter is in that object. It tells us how heavy or how light the object is.

mass The mass of an object is a measure of how much matter is in an object.

The SI unit for mass is the gram. A mass of one gram (1 g) is more or less equal to:

Other units of measurement used often are multiples of ten of 1 g. The most common multiples are shown in the table below.

### Worked example 7.3: Evaluating an estimation of mass

Danladi's mother sends his to the corner shop to buy a cup of rice. He carries it home in a small bag. He estimates that the mass of the rice is 50 g. Evaluate his estimation.

Step 1: Evaluate whether the unit of measurement is appropriate.

Think about how light 1 g is. It is about a quarter teaspoon of sugar. This is not a lot compared to a cup of rice.

It might be better to use kilograms, but then Danladi would need to use a decimal point in his estimation. To avoid the decimal point, gram is an appropriate unit.

Step 2: Evaluate whether the value of the estimation makes sense.

The mass of a teaspoon of sugar is about 4 g. So 50 g of sugar is = 12.5 teaspoons. Even though the same amount of sugar and rice have different masses, there is a big difference between 12.5 teaspoons and 1 cup.

Danladi's estimation does not make sense. An estimation of 150 g would be a better estimation.

### Exercise 7.5: Estimate mass of objects

Estimate the mass of the following objects. Include an appropriate unit of measurement in each case. Then use an electronic balance from the science laboratory or a kitchen scale to get the real measurements. Compare the real measurements to your estimations.

4. Full box of chalk
5. Volleyball

### Exercise 7.6: Estimate mass of everyday objects

Estimate the mass of the following objects by deciding which one of the options a) to d) is the most correct.

## Estimating and rounding

It is important to support students to use estimation to check the reasonableness of answers to calculations. In any situation involving a calculation the first thing to decide, given the context, is whether an estimate or an exact figure is required. Overwhelmingly adults use estimation rather than an exact calculation in everyday mathematics situations. It could be argued that in an increasingly cash less society, estimation is a vital skill. Developing strategies for estimating involves using thinking based on prior knowledge and therefore making connections with similar calculation situations. Estimation strategies can be applied at the beginning, end or part way through a calculation.

Rounding numbers makes numbers that are easier to work with mentally. Students should be able to use rounding in Number to the nearest 1, 10, 100, 100 etc. Front end estimation uses the number in the highest value place first and then, if necessary, the places immediately prior in order are considered. For example to find the approximate sum of 6554, 954 and 2676, the number in the thousands place are added to make 8 (thousand), then a glance at the numbers in the hundreds place will lead to providing for at least another thousand.

Similarly, to find the difference between 57,829 and 76,964, the 57 and 76 (thousands) are considered at the onset to result in about 9 (thousand), then the numbers in the hundreds place are referred to. In this case the estimate may remain at 9 thousand, depending on the accuracy the context requires. Approximating measurement to units of measure such as centimetre, minute, dollar etc is often useful for situations when exact amounts are not known. For example, &lsquoEach area is about a square metres and there are 7 areas, so that&rsquos about 7 square metres&rsquo &lsquoThree of the trips will take less than an hour and two will take just over an hour, so about five hours are needed&rsquo.

### Victorian Curriculum

Use estimation and rounding to check the reasonableness of answers to calculations (VCMNA182)

VCAA Sample Program: A set of sample programs covering the Victorian Curriculum Mathematics.

VCAA Mathematics glossary: A glossary compiled from subject-specific terminology found within the content descriptions of the Victorian Curriculum Mathematics.

##### Achievement standards

Students recall multiplication facts to 10 x 10 and related division facts. They choose appropriate strategies for calculations involving multiplication and division, with and without the use of digital technology, and estimate answers accurately enough for the context.

Students solve simple purchasing problems with and without the use of digital technology. They locate familiar fractions on a number line, recognise common equivalent fractions in familiar contexts and make connections between fractions and decimal notations up to two decimal places.

Students identify unknown quantities in number sentences. They use the properties of odd and even numbers and describe number patterns resulting from multiplication.

Students continue number sequences involving multiples of single-digit numbers and unit fractions, and locate them on a number line

## Estimation Worksheets

Estimation is a quick way of calculating the approximate answer to a math problem. Estimation helps people cross-check answers and find approximate values without spending too much time doing lengthy math problems. Teach your students this useful math skill with our free rounding and estimation worksheets. Kids can begin learning to estimate at a very young age. For kids who know how to add, use our estimating sums worksheets to teach them the basics of estimation. As kids learn how to multiply, they can also start practicing with estimating products worksheets as well.