Download Estimating with Decimals

2 .3
What are you going to
learn?
To estimate the sum
and the difference of
decimals
To estimate the
products of decimals
To estimate the
quotients of decimals
To add decimals
To subtract decimals
To multiply decimals
To divide decimals
Key Terms:
add, subtract, multiply,
divide, estimate, product,
quotient
Estimating with Decimals
1. Rounding to the nearest
Do you like ice cream?
Do you know what is in a cup of
ice cream?
Nutrition:
• Energy : 60.53 kal
• Protein : 1.938 gr
• Fat
: 0.58 gr
• Carbohydrate : 22.01 gr
C
Fi
You can round 60.53 to the nearest tenth. Think
about how you round the whole numbers. The rules
are similar.
EXAMPLE 1
Round 0.237 to the nearest tenth.
You can use a number line:
0.2
0.3
0.237
0.237 is closer to 0.2 than 0.3.
Therefore, to the
nearest tenth, 0.237 rounds to 0.2
Mathematics for Junior High School – Year 7/87
If you round 0.237 to the nearest hundredth, then
you get
0.23
0.24
0.237
0.237 is closer to 0.24 than 0.23.
Therefore, to the
nearest hundredth, 0.237 rounds to 0.24
You can round decimal the underlined place with
mental math.
0.237
2 is in the tenths’ place so 0.237 rounds to 0.2 or 0.3
3 is the digit to the right of 2
Since 3 < 5, 0.237 is closer to 0.2 than 0.3
To the nearest tenth, 0.237 rounds to 0.2
Problem 1 Round above to the nearest hundredth
Then round each decimal to the nearest tenth.
ROUNDING DECIMALS
Decide to which place you are rounding
• If the digit to the right of that place is greater
than or equal to 5, round up.
• If the digit to the right of that place is less than
5, round down.
88/Student’s Book – Operations of Decimals
2. Estimating Sums and Differences
2.1.
Estimating by Rounding
You can use rounding to estimate a sum or a
difference.
Example 2
Mother buys two fishes at the market. The first fish
is 0.426 kilogram and the second is 0.271 kilogram. If
the cost of 100 gram is Rp 1,000.00, estimate nearest
thousands mother will pay.
100 gram = 0.1 kilogram
nearest tenth
0.426
0.4
0.271
0.3
0.7
The weight of fish is about 0.7 kilogram. So, mother
will pay for about Rp 7,000.00.
Problem 2
Ms. Eni buys fish too. She chooses 0.738 kilogram
and 0.652 kilogram. If the price is also Rp 1,000.00
per 100 grams, estimate how many thousands Ms.
Eni must pay for the fish. If Ms. Eni wants to buy 2
kilograms of 3 fish, estimate the weight of the third
fish.
2.2.
Front-end estimation
Another type of estimation is front-end estimation.
When you use front-end estimation, add or subtract
the front digits. Then add or subtract the digits in the
Mathematics for Junior High School – Year 7/89
next place value position.
Example 3
Use front-end estimation to estimate the total weight
of three durians that are marked 2.193 kilograms,
1.507 kilograms, and 0.831 kilograms.
Add the front-end
digits, the kilograms
2.193
1.507
0.831
3
Estimates the
grams
0.193
0
0.507
1
0.831
1
2
Add
3
2
5
The total weight of the three durians is about 5
kilograms.
Use calculator to find the exact total weight of that
three durians.
How reasonable is the estimate?
Problem 3
Use front-end estimation to estimate the total weight
of the following items: apples 2.560 kilograms,
oranges 3.184 kilograms, mangoes 5.032 kilograms,
and grapes 1.593 kilograms. Check your answer with
a calculator. How reasonable is the estimate?
2.3.
Clustering
When estimating a sum in which all of the addends
are close to the same number, you can use clustering.
90/Student’s Book – Operations of Decimals
Example 4
Use the information in the table to estimate the total
number of hours worked in the four months.
Month
Hours Worked
May
72.50
June
68.50
July
69.75
August
71.75
The addends are clustered around 70.
72.50
70
68.50
70
69.75
70
+ 71.75
+ 70
280
So, a good estimate is 4 X 70, or 280. The total
number of hours worked in the four months is 280
hours.
Problem 4
A cage of rabbits at a pet store is given a vitamin
each day. The rabbits consumed 21.8 ounces, 19.1
ounces, 20.3 ounces, 18.9 ounces, and 20.3 ounces of
the vitamin each day. Use this information to
estimate the total amount of vitamin that is
consumed in one day.
Explain your answer.
Mathematics for Junior High School – Year 7/91
Concept
Summary
Front-End
Estimation
Estimation Methods
Estimate by first adding or
subtracting the front digits.
Then add or subtract the next
digits.
Estimate by rounding a group
of close numbers to the same
number
Clustering
3.
Estimating Products and Quotients
Buying Fruits. Dhani buys some apples at the
supermarket. The weight of the apples is 2.35 kg. If
the price of the apples is Rp 1,199.00 every 100
grams, estimate how many thousands Dhani will
pay for them.
Fruit market
To solve the problem above, try to answer the
following question.
1.
What two numbers will you multiply to find
the amount of money that Dhani should pay for the
apples?
2.
Estimate
the
price
that
Dhani
pays
by
rounding both numbers to the nearest thousand.
3.
Reasoning. How is your estimatimation
compared with the exact answer?
3.1. Estimating Products
You can estimate a product by rounding each factor
to the nearest whole number or by using compatible
numbers.
92/Student’s Book – Operations of Decimals
Example 5
Problem 5
Estimate
23.06 × 2.31
23.06
2.31
⎯⎯⎯×
⇒
⇒
23
Use compatible numbers
2
such as 23 and 2
⎯⎯×
46
The estimate product is 46.
Estimate the product 8.37 × 13.62.
3.2. Estimating Quotients
You can also use compatible numbers to estimate
quotient.s
Example 6
A stick is 92.8 cm long. A pencil is 15.6 cm long.
About how many times as long as the pencil is the
stick?
92.8 : 15.6
Write numerical expression.
90 : 15
Use compatible numbers to divide.
The stick is about six times as long as the pencil.
Problem 6
Use compatible numbers to estimate 68.2 : 7.52.
Adding and Subtracting Decimals
To add or subtract decimals, write the numbers in a
column
and line up the decimal points. In some
cases, you may want to place zeros at the back of the
decimals, to help align the columns. Then add or
Mathematics for Junior High School – Year 7/93
subtract in the same way as the whole numbers and
bring down the decimal point. Always estimate first
to determine whether your answer is reasonable.
Example 7
Find each sum or difference.
a. 12.65 + 4.5 Estimate 13 + 5 = 18
12.65
Line up the decimal points.
+ 4.50
Place a 0 to align the columns and add
17.1
The sum is close to the estimate.
b. 42.1 – 15.85 Estimate 42 – 16 = 26
42.10
- 15.85
26.25
Problem 7
Place a 0 and line up the decimals points
Subtract
The difference is close to the estimate.
Find each sum or difference.
a.
57.125 + 7.63
b.
25 – 15.25
Multiplying and Dividing
Decimals
Multiplying Decimals
Consider the following problem: Six tables are lined
up end-to-end. Each table is 2.3 metres long. How
long is the line of the tables?
The
problem
can
be
multiplication problem.
94/Student’s Book – Operations of Decimals
seen
as
the
following
6 × 2.3 with the estimate 6 × 2 = 12
The exact answer is:
6 × 2.3 = 2.3 + 2.3 + 2.3 + 2.3 + 2.3 + 2.3 = 13.8
Which is reasonably close to the estimate.
Instead of using repeated addition as above, you can
think of the following strategy.
2.3
X
23
6
tenth (1 decimal place)
6
X
138 tenth, which is equal to 13.8
(1 decimal place)
Now,
using
the
similar
strategy
we
multiply
decimals.
Example 8
Multiply 1.4 × 0.9 Estimate
1.4
× 0.9
1.5 × 1 = 1.5
14
tenth (1 decimal place)
× 9
tenth (1 decimal place)
126
hundredth, which is equal to 1.26
(2 decimal places).
Compare the result with the estimate.
Problem 8
Multiply 0.84 × 1.6
1.
Dividing Decimals
Consider the following problem.
A gasoline company distributed 97.2 million litres of
gasoline equally to six costumers. How much
gasoline did each customer receive?
Did each
costumer get more than fifteen million litres? Did
Mathematics for Junior High School – Year 7/95
each costumer get more than twenty million litres?
What is the answer?
The problem is a division problem of the following:
97.2 : 6
16.2
6 97.2
- 60.0
37.2
- 36.0
1.2
- 1.2
0
with the estimate 90 : 6 = 15
Place 0s to align the columns then subtract
Place 0 to align the columns then subtract
Divide 12 tenths to 6 people, each gets 2 tenths
Thus, 6 × 2 tenths or 12 tenths have been
distributed .
Compare the result with the estimate.
Example 9
Divide 97.5 : 6
The estimate is 90 : 6 = 15
The problem can also be seen as 97.50 : 6
16.25
6 97.50
- 60.00
Place 0s to align the columns then subtract.
37.50
- 36.00
1.50
-
1.20
0
Place 0 to align the columns then subtract.
Divide 12 tenths to 6 people, each gets 2 tenths.
Thus, 6 × 2 tenths or 12 tenths have been
distributed.
Compare the result with the estimate.
96/Student’s Book – Operations of Decimals
To divide a decimal by decimal you can turn the
problem by making the divisor as a whole number.
Then follow the algorithm above.
To make the divisor as a whole number means you
should multiply the divisor by a power of ten, such
as 10, 100, 1000.
In order not to change the problem, the dividend
must also be multiplied by that same number. For
example:
0.5 1.25
convert to
5 12.5
Multiply by 10
Problem 9
Divide 199.68 : 9.6
Concept summary
To multiply decimals, multiply them in the same
way as whole numbers. The product has the same
number of decimal places as the sum of the
decimal places of the factors. Use estimate to
determine whether your answers are reasonable.
To divide two decimals, use the following:
• If necessary, change the divisor to a whole
number by multiplying the divisor by a
power of ten.
• Multiplying the dividend by the same power
of ten.
• Divide in the same way as whole numbers.
Mathematics for Junior High School – Year 7/97
1. Round each decimal to the nearest hundredth.
Then round each decimal to the nearest tenth.
a. 3.4726
b. 0.5873
c. 2.095
d. 0.053
e. 0.6853
f. 42.063
g. 1.001
h. 123.456
2. Open-ended . Write five different decimals that
are rounded to 5.7
3. Round each number to the underlined place.
a. 3.166
b. 0.428
c. 3.2197
d. 0.0806
4. Estimate each sum or difference to the nearest
integer. Then compare the estimate with the exact
results.
a. 3.89
b. 4.521
c. 0.82
d. 23.45
+ 2.92
- 1.938
+ 3.39
- 12.97
e. 12.67
f. 0.49
g. 32.51
h. 5.55
- 9.52
+ 2.38
- 27.99
- 4.44
5. Round each factor to the nearest whole number to
estimate the product. Then compare each estimate
with the corresponding exact product.
a. 17.2 × 1.8
b. 3.3 × 12.6
c. 0.67 × 35.4
d. 23.9 × 9.38
e. 35.04 × 7.82
f. 47.3 × 0.82
g. 0.38 × 0.51
h. 49.2 × 3.81
6. Write a pair of compatible numbers. Then
estimate each product.
a. 73.6 × 1.48 b. 56.39 × 3.053
98/Student’s Book – Operations of Decimals
c. 0.41 × 31.06
d. 0.7 × 1.53
g. 2.74 × 1.837
e. 73 × 1.65
f. 0.06 × 42.49
h. 567.2 × 11.39
7. Use compatible numbers to estimate each
quotients.
a. 64.4 : 536
b. 32.7 : 8.06
c. 8.03 : 0.52
d. 37.45 : 4.06
e. 53.8 : 23.7
f. 0.76 : 0.03 30.
g. 36.7 : 6.45
h. 5.61 : 27.9
8. Determine two integers that are close to the
quotient 28.4 : 3.58.
9. Complete the table below.
No
a.
b.
c.
Sum
Estimate to
the nearest
integer by
using
clustering
Exact
result
Difference
6.99
+6.59 +
7.02 +
7.44
5.45 +
5.3948
+
4.7999
99.8 +
100.2 +
99.5
+100.4
10. Number Sense. Estimating 26.25 : 4.62.
a. First use the compatible numbers.
b. Then estimate by rounding.
c. Which method is closer to the exact answer?
Mathematics for Junior High School – Year 7/99
Why ?
11. Consumer Issues. Regular gasoline costs Rp
4.800,00/litre. Mr. Bambang spends Rp 15,000.00
on gasoline. How many litres does he buy?
12. Reasoning. Is an estimate of 20 higher or lower
than the sum of 6.66, 9.54 and 4.813? How can
you tell? Is the estimate reasonable? Explain.
13. Writing. Describe a situation in which you might
want your estimate to be high. Then describe one
in which you might want your estimate to be low.
14. Research. Look through newspapers or magazines
to find five decimals. Decide whether each
decimal is an estimate. To which places are the
decimals rounded?
15. Number sense. How do you know that the sum of
5.4, 6.3, and 9.6 is greater than 20?
16. Nutrition. Use the chart on the left.
Vitamin B (mg) in 100
grams
Heart
9.25
Liver
0.53
Tongue
0.12
Brain
0.23
Small
0.08
Intestine
Intestine
0.15
Estimate and round to the nearest tenth.
a. About how much vitamin B is in heart and
tongue? In one of everything?
b. About how much more vitamin B is in 500
gram of liver than that in 1 kilogram of brain?
c. About how much vitamin B is in the last three
items combined?
17. Mental Math. Suppose you use a hand phone for
3.07 minutes. If the calling cost is Rp 275.47 per
minute, estimate how many thousands you will pay
for it.
100/Student’s Book – Operations of Decimals
18.
Mental Math. Suppose you save Rp 3,870.00
each week. Estimate how much money you will save
in one year.
19. Social Studies. The Apennine Railroad Tunnel
in Italy is 18.5 km long. The Seikan Tunnel in
Japan is 2.9 times longer. Estimate the length of
the Seikan Tunnel.
20. Packaging. A volley ball weighs 283.5 gram and
a shipping crate weighs 595.34 gram. Estimate the
weight of a shipping crate containing 9 volley
balls.
21. Nutrition. Use the chart below to answer.
Food
Serving Size
Canned Tuna 3 oz (drained)
Rye Bread
1 slice
Cheese Pizza 1 slice (14 in.
pie)
Protein (grams)
24.4
2.3
7.8
a. About how many grams of protein are in 2
slices of cheese pizza?
b.
About how many grams of protein are in 8
slices of cheese pizza?
c. Estimate how many grams of protein are in a
sandwich consisting of 2 slices of rye cheese
bread and 2 oz of tuna.
22. Library Science. The bills for 3 copies of a book
is Rp 38.840,00. Estimate the cost of one book. Is
Mathematics for Junior High School – Year 7/101
your estimate higher or lower than the actual
cost of the book?
102/Student’s Book – Operations of Decimals