Lesson 8
Practice Lesson 8 Solve Problems with Rational Numbers
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Lesson 8
Solve Problems with Rational Numbers
Name:
Prerequisite: Add Negative Fractions
Study the example showing how to add negative fractions. Then solve problems 1?8.
Example
Marita
has
a
container
of
oatmeal.
She
uses
1 ?4?
cup
of
the
oatmeal
for
breakfast
and
3 ?8?
cup
for
a
cookie
recipe.
What
is
the total change in the amount of oatmeal she has?
You can use a number line to represent this problem. Keep in
mind
that
1 ?4?
is
equivalent
to
?82?.
2
3 8
2
1 4
22
2148
21
2
4 8
0
B
1
Why
are
1 ?4?
and
3 ?8?
represented
with
negative
numbers
on the number line in the example?
1 ?4?
and
3 ?8?
are
represented
with
negative
numbers
on
the
number
line
because
Marita
uses
1 ?4?
cup
and
3 ?8?
cup
of
oatmeal,
so
she
is
taking
these
amounts
out
of
the
container.
B
2
What do line that
you need to is measured
do to 2?14? to in eighths?
locate it Rewrite
on a number 2?14? as 2?28?.
B 3 You can also use an equation to solve the problem.
Complete the addition equation.
Change in oatmeal after breakfast 1
1 2?41? 2
1
Change in oatmeal after cookie recipe 5
1 2?38? 2
5
Total change ?
B 4 What is the total change in the number of
cups in the oatmeal container? Explain.
2?58?;
1
2?41?
2
1
1
2?38?
2
5
2?28?
2
3 ?8?
5
2?85?
cup
of
oatmeal
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Lesson 8 Solve Problems with Rational Numbers
6699
Solve.
M
5
Nien
has
a
flower
farm.
He
cuts
4
3 ?4?
square
yards
of
flowers
one
day
and
5
2 ?3?
square
yards
of
flowers
the
next.
Use addition to find the total change in the square yards
of flowers.
Show your work.
1
24
3 ?4?
2
1
1
25
2 ?3?
2
5
24
9 ?1?2?
2
5
8 ?1?2?
5
29
17 ?1?2?
5
210
5 ?1?2?
Solution:
The
total
change
is
210
5 ?1?2?
square
yards.
B 6 Carl has a new bottle of laundry detergent. He uses
0.4 cup of laundry detergent while washing one load of laundry and 0.7 cup on another load. Use addition to find the total change in the number of cups of detergent in the bottle.
20.4 1 (20.7) 5 21.1 cups
M
7
What
is
the
sum
27
1 ?6?
1
3
5 ?6?
in
simplest
terms?
Show your work.
27
1 ?6?
1
3
5 ?6?
5
26
7 ?6?
1
3
5 ?6?
5
23
2 ?6?
5
23
1 ?3?
Solution:
23
1 ?3?
C 8 At a recycling center, items are sorted and removed
according to the material they are made out of. In one mixture of recycled materials, 2.625 tons of metal cans, 5.58 tons of paper, and some glass items are removed. If the total change in weight of the recycled materials is 29.5 tons, how many tons of glass were removed?
Show your work. Possible work: 22.625 1 (25.58) 5 28.205
change in tons of glass 1 (28.205) 5 29.5
29.5 2 (28.205) 5 29.5 1 8.205 5 21.295
7700
Solution: 1.295 tons of glass were removed.
Lesson 8 Solve Problems with Rational Numbers
?Curriculum Associates, LLC Copying is not permitted.
Unit 1 The Number System
Unit 1
B Basic
Key
M Medium
C Challenge
27
Unit 1 The Number System ?Curriculum Associates, LLC Copying is not permitted.
Practice and Problem Solving
28
Lesson 8
Name:
Estimating with Decimals
Study the example showing how to use estimation when computing with positive and negative decimals. Then solve problems 1?9.
Example
The temperature at noon was 5.6?F. During the afternoon the temperature dropped 2.8?F and then dropped another 2.2?F in the evening. Estimate the new temperature.
You can use a number line to understand the problem by approximating, or rounding, each temperature to the nearest degree.
5.6 rounds up to 6. 2.8 rounds up to 3. 2.2 rounds down to 2.
22 21 0 1 2 3 4 5 6 7
Now, you can estimate the new temperature by finding 6 2 3 2 2. So, using estimation, the new temperature will be about 1?F.
B 1 Based on the estimate, do you think the actual new
temperature will be above or below 0?F? Explain.
above; The estimate is 1?F, so the actual new temperature should be just a little above 0?F.
B 2 Write and solve an equation to find the new temperature
using exact values. What is the exact value of the new temperature?
5.6 2 2.8 2 2.2 5 0.6; 0.6?F
B 3 Peter got an exact new temperature of 11.6?F. Explain
how Peter can use the estimate to determine that his answer is wrong. Then find and correct Peter's error.
Peter's answer is much higher than the estimate. It should be pretty close to the estimate.
Peter added 2.8 and 2.2 to 5.6. He should have subtracted 2.8 and 2.2 from 5.6.
M 4 Janet estimated to the nearest half degree before she
found her exact answer. Is Janet's estimate more or less accurate than the estimate in the example problem? Explain.
more; In general, the more precise that the estimate of each temperature is, the more
accurate the estimate will be.
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Lesson 8 Solve Problems with Rational Numbers
7711
Solve.
M 5 Kyle and Joan are estimating the total weight of three
boxes that each weigh 14.62 pounds. Each student's estimate is shown below.
Kyle 3(15) 5 45 lb
Joan 3(14.5) 5 43.5 lb
Which estimate is more accurate? Explain how you know. Then find the actual total weight of the boxes.
Joan's estimate should be more accurate because she used a more precise estimate;
3(14.62) 5 43.86 lb
M 6 Estimate 55.8 4 (23.1). Then find the exact quotient.
Show your work. Possible work: estimate: 56 4 (24) 5 214 exact: 55.8 4 (23.1) 5 218 Solution: The estimate is 214, and the exact quotient is 218.
Use the following situation to solve problems 7?9.
Dusan had $49.60 in his checking account. He wrote one check for $24.40 and another check for $25.49.
M 7 Estimate the amount of money in Dusan's account after
he wrote the checks. Possible answers: $50 2 $24 2 $25 5 $1 or $50 2 $25 2 $25 5 $0
M 8 Explain why the estimate might not be enough
information to show Dusan whether he has enough money in his checking account to cover the checks.
The estimate shows that Dusan has very little in his account, but it is not accurate enough
to say for sure if he has enough in his account to cover the checks.
C 9 Dusan says that because he always rounds to the nearest
dollar, he will always have more money in his account than his estimate. Is Dusan correct? Explain.
No; rounding to the nearest dollar does not mean that all numbers will be rounded up.
7722
Lesson 8 Solve Problems with Rational Numbers
?Curriculum Associates, LLC Copying is not permitted.
Unit 1
Practice Lesson 8 Solve Problems with Rational Numbers
Practice and Problem Solving ?Curriculum Associates, LLC Copying is not permitted.
29
Unit 1 The Number System
Lesson 8
Name:
Estimating with Fractions
Study the example showing how to use estimation when computing with positive and negative fractions. Then solve problems 1?6.
Example
A
high
diving
board
is
about
9
5 ?6?
feet
above
the
water's
surface. A diver jumps off the diving board and ends up
3
1 ?4?
feet
below
the
surface
of
the
water.
What
is
the
total
distance of the jump?
You can use rounding and a number line to help you estimate the answer.
Actual Distance
9
5 ?6?
23?41?
Nearest Integer 10 23
24 23 22 21 0 1 2 3 4 5 6 7 8 9 10
B 1 To the nearest integer, what is the distance of the jump?
Based on the number line, the distance is about 13 feet.
B 2 What is the exact distance of the jump?
Show your work. To find the distance, I'd add the distance of each point from 0.
9
5 ?6?
1
3
1 ?4?
5
9
10 ?1?2?
1
3
3 ?1?2?
5
12
13 ?1?2?
5
13
1 ?1?2?
Solution:
The
diver
jumped
a
distance
of
13
1 ?1?2?
feet.
B 3 Compare the estimated distance of the jump to the actual
distance? Is the actual distance reasonable?
The distances are close, so the actual distance is reasonable.
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Lesson 8 Solve Problems with Rational Numbers
7733
Solve.
M
4
Think
about
the
problem
26
3 ?5?
2
1
27
?145?2
1
2
?51?.
a. Estimate the answer.
27 2 (27) 1 2 5 27 1 7 1 2 5 2
b. Find the actual answer.
26
3 ?5 ?
2
1
27
4 ?1?5?
2
1
2
1 ?5?
5
26
9 ?1?5?
1
7
4 ?1?5?
1
2
3 ?1?5?
5
2
13 ?1?5?
c. Compare the estimate and the actual answer. Is your actual answer reasonable?
The actual answer is close to the estimate, so the actual answer is reasonable.
M
5
James
baked
5
1 ?4?
dozen
peanut
butter
cookies
and
4
5 ?6?
dozen oatmeal cookies for a bake sale.
a. Estimate the number of dozens of cookies he baked.
5 1 5 5 10 dozen
b. Find the actual number of dozens of cookies he baked.
5
1 ?4?
1
4
5 ?6?
5
5
3 ?1?2?
1
4
10 ?1?2?
5
10
1 ?1?2?
c. How do the numbers compare? Is the actual number of dozens you found reasonable?
The actual number is close to the estimate, so the actual number is reasonable.
C 6 What is the smallest integer that makes the difference
x
2
1
3 ?4?
1
4 ?5?
2
greater
than
1?
Use
estimation
to
find
the
answer, and then find the exact difference to show that
your answer is correct.
Possible
answer:
Both
3 ?4?
and
4 ?5?
round
to
1,
so
the
sum
is
approximately
equal
to
x 2 (1 1 1), or x 2 2. Because 2 2 2 5 0 and 3 2 2 5 1, I know 3 would be the smallest
integer that would give a sum greater than 1.
3
2
1
3 ?4?
1
4 ?5?
2
5
3
2
1
15 ?2?0?
1
16 ?2?0?
2
5
1
?29?0?.
Because
1
9 ?2?0?
is
greater
than
1,
I
know
that
3
is
the
makes the difference greater than 1.
7744
Lesson 8 Solve Problems with Rational Numbers
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Unit 1
Practice Lesson 8 Solve Problems with Rational Numbers
30
Practice Lesson 8 Solve Problems with Rational Numbers
Practice and Problem Solving
Lesson 8
Solve Problems with Rational Numbers
Solve the problems.
Name:
B
1 Which expression is the best estimate of 1.25 1 (22.69)?
How might a number
A 1 1 (23)
C 2 1 (23)
line help you
B 1 1 (22)
D 2 1 (22)
estimate decimals?
Sue chose B as the correct answer. How did she get that answer?
Sue estimated 22.69 as 22 instead of 23, which is
actually closer to 22.69 on the number line.
B
2
Which
expression
is
the
best
estimate
of
1
1 ?5?
3
3
?32??
A 234
B 134
C 233
D 133
Should you round each value up or down?
M
3 The answer to which situation is best estimated
by 27?
A The temperature falls 1.9?F from 4.3?F.
B A hot-air balloon rises 4.1 meters from the ground and then rises 2.3 meters more.
How can you estimate decimal amounts?
C A submarine goes 3.8 feet below the surface of the ocean and then goes 2.5 feet farther down.
D The amount of money Sarah has after she buys milk for $3.85 and bread for $2.45.
Solve.
M
4 Tell whether each statement is True or False.
a.
23
3 ?4?
1
1
5
23
2 ?3?
2
(21)
b. 2.855 2 1.375 5 1.48
u u3 True
False
u3 u True
False
c.
212
7 ?12?
1
1
29
5 ?6?
2
5
3
7 ?12?
d.
6
2 ?5?
2
1
27
1 ?10?
2
5
13
1 ?2?
u u3 True
False
u3 u True
False
Can you tell if the sum or difference will be positive or negative?
M
5 Justin earned $25.67 one week and $37.85 the next
week. He also purchased a DVD for $19.99 and gas for
his car for $22.07. He started with a zero balance in his
checking account. Justin recorded all of these
transactions in his checkbook and ended up with a
An estimate might help you determine whether Justin is correct.
balance of $105.58. Was his balance correct? If it was,
write an equation to support the answer. If not, find the
correct balance.
Justin is not correct. Justin should have subtracted
his total purchases (19.99 1 22.07 5 42.06) from
his earnings (25.67 1 37.85 5 63.52). His balance is
$63.52 2 $42.06 5 $21.46
C
6 Write a sum, difference, product, and quotient whose
answers could all have an estimate of about 6. One expression should include only mixed numbers, one expression should include only decimals, one expression should include only negative rational numbers, and one expression should include only positive rational numbers.
When do you round a rational number up? When do you round a rational number down?
Sum: Possible answer: 1?43? 1 3?56?
Difference: Possible answer: 12.2 2 5.9
Product: Possible answer: 22.9(22.1)
Quotient: Possible answer: 18.1 4 3.2
Unit 1 The Number System ?Curriculum Associates, LLC Copying is not permitted.
?Curriculum Associates, LLC Copying is not permitted.
Lesson 8 Solve Problems with Rational Numbers
7755
7766
Lesson 8 Solve Problems with Rational Numbers
?Curriculum Associates, LLC Copying is not permitted.
Unit 1
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