UNIT THREE: Prealgebra in a Technical World
3.8 Using Calculators
SWBAT 1. Use calculators correctly.
2. Evaluate formulas and complete long calculations.
Calculators do not keep any of us from making mistakes. If we are not thinking, we
can easily make mistakes with a calculator and never know we have made an error! We still
estimate and stay reasonable to solve problems successfully when using technology.
Estimating results by rounding, calculating mentally, using rules of signs, and recording
our thinking step by step — all of these are the basic skills we need to stay accurate as we use
calculating technology.
In this section we study using hand-held calculators successfully.
Using Calculators
1
1
Scott ran 2 3 miles on Monday and another 4 6 miles on Tuesday. First he estimates
that his average is about 3 miles per day. He gets out his calculator to find the exact answer.
Scott enters “2 + 1 ÷ 3 + 4 + 1 ÷ 6 ÷ 2 = ” and his calculator evaluates this: “6.75.”
Scott knows this answer must be wrong. Scott does not know why his calculator is not working
for him, so he takes a different approach.
Scott adds mentally, “2 + 4 = 6, and
1
3
+
1
6
3
𝟏
1
6
𝟐
2
= = so the sum is 6 .” He uses his
1
calculator to divide 6 2 by 2. Scott enters “6 + 1 ÷ 2 ÷ 2 =” into his calculator. The calculator
evaluates this as “6.25.” This is still too large. Scott gives up and gets out a paper and pencil.
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SECTION 3.8 Using Calculators
Scott is making a very common error. Fraction bars are grouping symbols, but
calculators cannot see numerators and denominators. To tell the calculator to
separate numerators and denominators, we need to wrap our numerators and
denominators in separate parentheses.
Scott needed to enter "(6 + 1 ÷ 2)/2 = . " While Scott would do better to simply use
mental math for this calculation, sometimes we will want to use calculators to take the
drudgery out of computing long fraction problems.
To show symbols and the order of entering expressions into scientific calculators, we
use a key sequence.
DEFINITION: A key sequence is the list of key strokes in the order that they are
entered in a calculator. In writing a key sequence, we put boxes around all keys
that are not one of the ten digits. For the digits 0 to 9, we simply write the digit.
1
To find Scott’s average running distance;
1
23 + 46
2
, we use the following key sequence
read from left to right; (.. 2 .+. 1 .÷. 3 .+. 4 .+. 1 .÷. ..6 ..). .÷. 2 .=. . In this key
sequence we need parentheses around the whole numerator, but order of operations takes
care of adding the mixed numbers in that numerator, because division comes before addition.
1
Scott’s calculator evaluates this key sequence as 3.25, and Scott knows that this is 3 .
4
1
This result fits his estimate. On average Scott ran 3 4 miles per day.
Example 1: Simplify this fraction using a calculator
4+2∙5
5+6÷3
Think it through: We use a key stroke guide to show how to enter this expression in a
scientific calculator:
.(.. 4 .+. 2 x 5 ..). .÷. .(.. 5 .+. 6 .÷. 3 ..). .=.
ANSWER:
𝟒+𝟐∙𝟓
𝟓+𝟔÷𝟑
=2
UNIT THREE: Prealgebra in a Technical World

Check Point 1
a) Write the key sequence for 3 +
2+5∙4
11
b) Simplify with your calculator and check this result using your estimate or mental math.
At times you will want to use the opposite key. On some calculators it is a dash; on
others, it is has both a plus and a dash. The key is just to the left of your equal sign. The two
symbols used most often to indicate the opposite are:
+/-.
and
(-) .
Another valuable key is the exponent key. This key is often .^.. (called the carat key)
or . xy. on your calculator. Often the exponent key is on the right side of the calculator, just
above the other operations.
To get a feel for these, we review the sign rules for exponents, but this time we use a
calculator.
Example 2: Use your calculator to complete the following. (1) Write the key sequence.
(2) Write the result.
2 2
2 2
2 3
b. − ( )
a. (− )
3
2 3
d. − ( )
c. (− )
3
3
3
ANSWER:
a.
(- (-) 2 .÷ 3 )
^
2 .=. The result is +
b.
(-) ( 2 .÷ 3 )
^
2 .=. The result is −
c.
(
^
3 .=. The result is −
d.
(-) ( ) 2 .÷ 3 )
(-) 2 .÷ 3 )
^ 3 .=.
The result is
−
4
9
4
9
8
27
8
27
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272
SECTION 3.8 Using Calculators

Check Point 2
Use your calculator to complete the following. Key in carefully. Write the result.
4 2
1 4
2 2
2 4
a. − ( )
b. (− )
c. (− )
d. − ( )
a. _________
b. __________
c. _________
d. _________
5
2
5
3
Many calculators have a fraction key or a fraction function on the menu. This key will
change the decimal for your fraction remainder into a fraction. If you have one, your fraction
key may look like a b/cc
,
F , F. , or FD , and any of these symbols might be
above a key. This means that you would have to use a .2nd. function key to choose this
operation. Some calculators have the fraction feature on the MATH menu.
If you have trouble using your calculator, help is available. Your instructor, your friends,
fellow students, and the tutors in the Tutoring Center can help you find and use calculator
functions. For any feature of your calculator, you may also be able to read your user’s manual
or go online for instructions.
You do not need a calculator with a fraction key for this class. If your calculator does
not have a fraction key, you can learn to recognize the decimal equivalents for many fractions.
Half Thirds
0.5
Fourths Fifths
Sixths Sevenths
Eighths
̅̅̅
0. ̅33
0.25
0.2,
0.16̅
0. 142857
0.125
̅̅̅
0. ̅11
0.1
̅̅̅̅
0. 66
0.75
0.4,
0.83̅
0. 285714
0.375
̅̅̅̅
0. 22
0.3
0.625
̅̅̅̅
0. 44
0.5
0.875
̅̅̅̅
0. 55
0.7
̅̅̅
0. ̅77
0.9
0.6,
0.8
0. 428571
0. 571428
0. 714285
0. 857142
Ninths
̅̅̅
0. ̅88
Tenths Elevenths
̅̅̅̅
0.0909
̅̅̅̅
0.1818
̅̅̅̅
0.2727
̅̅̅̅
0.3636
̅̅̅̅
0.4545
̅̅̅̅
0.5454
̅̅̅̅
0.6363
̅̅̅̅
0.7272
̅̅̅̅
0.8181
̅̅̅̅
0.9090
UNIT THREE: Prealgebra in a Technical World
Evaluating Formulas and Completing Long Calculations
Once you have mastered your keys and key strokes, you are ready to add the calculator
to your set of problem-solving skills. Calculators do not replace estimating, using mental math
or writing down the steps we take. Calculators can simplify computations, but you use them
successfully only when you know your math!
Example 3: On July 30, 2009, temperature records were broken all over the Pacific Northwest.
Seattle, WA, had its hottest day in recorded history. The thermometer rose to 103°F on that
day. What was the high temperature in degrees Celsius for Seattle that day?
𝟓
Think it through: We use the formula 𝑪 = 𝟗 (𝑭 − 𝟑𝟐) with 𝑭 = 𝟏𝟎𝟑°
𝟏
Estimate using 𝐶 = (103 − 30) ÷ 2. 103 − 30 is 73. And 73 ÷ 2 is 𝟑𝟔
𝟐
5
Write the formula with 103 for F. 𝐶 = (103 − 32)
9
5
Subtract inside the parentheses. 𝐶 = (71)
9
Multiply 5 and 71, then divide by 9.
Or use this key sequence: 5 .x. 7 1 ÷. 9 .=.
Write the mixed number answer. 39
4
9
4
Note: If you divide by hand, you have the mixed number 39 9. On a calculator use the
fraction key to change your decimal, 39.44444, to a mixed number. Better yet learn to
recognize the decimal representation of fractions like those on the last page.
𝟒
ANSWER: The high temperature for July 30, 2009, in Seattle, WA, was 𝟑𝟗 𝟗 °𝑪.
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SECTION 3.8 Using Calculators

Check Point 3
Use the formula to find the Celsius temperature for Medford’s July 30, 2009, high of 109°F.
Write your answer as a fraction.
While calculators are the most useful tools we have for taking the drudgery out of math,
when we misuse calculators by trying to get them to think for us, they make math calculations
harder than they need to be and they produce results that are impossible to check.
Often it is easier to take many of the steps needed to calculate a result by using mental
math.
Example 4: Just like the water in a river, the electricity flowing through wires slows down as it
meets with the resistance inherent in all wires. We see this slow down as a decrease in voltage.
Dave, who is majoring in electronics, is planning to use a compressor as he works on his
travel trailer in the back yard. Before he does, he figures out whether he can get enough volts
with the extension cord he is using. Dave’s compressor must have at least 100 volts to run.
The formula he uses is 𝑉 = 120 − 2 ∙ 𝐴 ∙ (𝑅) ∙ (𝐿) where 𝑉 is the delivered voltage, 𝐴 is
the amperage of Dave’s compressor, 𝑅 is the resistance in the wire, and 𝐿 is the length of the
cord. Dave’s compressor has amperage of 15, the resistance for the extension cord is 3/439
and his extension cord is 100 ft long. Will there be enough voltage to run the compressor?
UNIT THREE: Prealgebra in a Technical World
Think it through:
 Use the formula 𝑽 = 𝟏𝟐𝟎 − 𝟐 ∙ 𝑨 ∙ (𝑹) ∙ (𝑳) for = 𝟏𝟓 , 𝑹 = 𝟑/𝟒𝟑𝟗 and L = 𝟏𝟎𝟎 .
 Dave estimates that his voltage will be close to 100. This one is too close to call!
3
𝑉 = 120 − 2 ∙ 15 ∙ (439) ∙ (100) Dave substitutes the correct values into the
formula.
3
= 120 − 2 ∙ 15 ∙ (
) ∙ (100) He uses the commutative property to multiply
439
the integers 2 ∙ 15 ∙ 100.
3
3
= 120 − 3,000 ∙ (
)
439 He keys 3,000 ∙ (439) into the calculator using
this sequence:
(Dave’s calculator reads: 20.50113895.)
120+ ≈ (−20.5) < 100 Dave notes his calculator result, and sees that
120 plus this number is less than 100.
ANSWER: With this extension cord, Dave knows he will not be able to operate his
compressor because the delivered voltage is below 100 volts. He must
invest in a different extension cord.
When we use long formulas or simplify long expressions, we need to keep our work
organized. At the same time we need to work efficiently just to get the job done in a reasonable
amount of time. Calculators help tremendously, but if we try to work by just using a calculator,
it is easy to make a mistake when entering symbols and numbers. These mistakes are not hard
to find if you have kept an organized record. You reduce your chances of making mistakes
when you use mental math as much as you can.
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SECTION 3.8 Using Calculators
Steps for Long Calculations
 Estimate as soon as possible and re-estimate along the way.
 Write down the whole formula or long expression at the top of your long calculation.
 Use order of operations and the properties to simplify at each step.
 Keep track of steps by writing them down as needed to keep organized. (This is often
called "showing your work." This is your work; show what you need to show. If it is work for
the boss, show what the boss needs to see.)
 Use as much mental math as you can. Mental math results are more reliable than those on
a calculator—we can hit the wrong key on a calculator and find a terribly wrong result for a
very easy calculation.
 Use a calculator when it saves time. Do not use a calculator when you can compute more
quickly without one. With exercise, your brain will perform amazing calculations without
the aid of paper, pencil or calculator.
 Check your work by reading your steps and comparing your answer to your estimate.
Sometimes you might even solve your problem a different way. Often you will think of an
easier way to solve after you have finished the whole problem.
 Make sure you answer the question! It is no fun to have all of the math done correctly, but
to forget to answer the question!

Check Point 4
Find the answer to this question and show your work in an organized way.
On January 16, 2009, the lowest temperature ever recorded in Maine was in Big Black
River, Maine, at -50°F. What is the Celsius reading for this temperature?
We practice our skills at working long calculations using exercises. These are not real
problems, just like football practice is not a real game. The real games, and the real problems,
UNIT THREE: Prealgebra in a Technical World
require that many more skills come into play all at once. By practicing long, and sometimes
tricky, calculation exercises, you focus on honing your calculating skills. When the real problem
(like the real game) arrives, your skills are ready to go.
Example 5: Simplify
3 −5 ∙ 2
8
5
Think it through: Show each step.
3 − 5 ∙ 2 Write the expression on paper.
8
5
(3 − 5 ∙ 2) ÷
8 Rewrite by dividing by the denominator.
5
. . . and
(3 − 10) ÷
8 Multiply within the parentheses.
5
−7 ÷
8 Subtract within the parentheses.
5
−7 5 Change division to multiplication by the reciprocal. Write
∙
1 8 the integer as a fraction with a denominator of 1.
−35
8
ANSWER:
3 −5 ∙ 2
8
5
= −4
In your algebra class, this is an acceptable answer. In prealgebra we answer with a mixed number.
3
8
We can give either the improper fraction or mixed number answer for this problem
depending on the application. We read the instructions or pay attention to what is appropriate
in other classes, at home, and on the job to know how to write the answer.
The odd problems in this section are intended to help you practice using mental math,
paper and pencil, and your calculator together. You will learn to calculate accurately while you
write the steps you use. As you practice, always estimate and be reasonable. While your
calculator ends the drudgery of long calculations, you are the one responsible for knowing that
your final answer is correct.
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SECTION 3.8 Using Calculators
Final message:
Do not be a slave to technology—become the master of your calculating machines!
UNIT THREE: Prealgebra in a Technical World
3.8 Exercise Set
Name _______________________________
Skills
Write the fractional equivalent or mixed number for the following calculator displays:
Calculator Display
Fraction
Calculator Display
1.
̅̅ or 1.333 …
1. ̅̅
33
2.
5.25
3.
21.125
4.
̅̅
4.1111 … or 4. ̅̅
11
5.
̅̅̅̅̅ …
2.16̅ or 2.1666
6.
11.6666 …
7.
5.8
8.
7.375
9.
3.83̅ or 3. 8333 …
10.
̅̅̅ or 11.111 …
11. ̅11
11.
1. ̅̅̅̅̅̅̅̅̅̅
142857
12.
52.875
13.
̅̅̅̅
6.4545
14.
̅̅̅
12. ̅88
Fraction
Use your calculator for the following. (1) Write the key sequence. (2) Write the result.
15.
16.
17.
18.
19.
20.
21.
Problem
9−2∙3
22
42 − 1
23
14
32
7
3
1
4+8
2
−52 − 2
32
5
4−3÷
6
9 − 15
12 − 3 ∙ 22
Key Sequence
Result
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280
SECTION 3.8 Using Calculators
22.
3 1
22 ∙ 3
4 2
1 2
(5 )
23.
3
3
1
1
24. ( 4 ) ∙ (− 2) ÷ 4
25.
26.
3 2
− ( 4)
3 2
(− 4)
3
3
2 −5
8
4
Use your calculator and formulas to find the AREA. Use your 𝜋 key when needed.
If you do not recognize the fraction, round to the nearest one hundredth.
5
3
27.
a triangle with a height of 21 8 inches and a base of 22 4 inches
28.
a triangle with a base of 22 feet 4 inches and a height of 14 feet 8 inches
29.
a square with a side of 9 10 meters
30.
a square with a side of 4 feet 8 inches
31.
a rectangle with sides 12 feet 3 inches and 10 feet 6 inches
32.
a rectangle with sides 4 1/2 miles by 7 3/4 miles
33.
a parallelogram with base of 56 feet, 9 inches and height of 30 feet, 3 inches
34.
a parallelogram with base of 2 meters and a height of 1/4 meters
35.
a trapezoid with bases 15 3/4 feet and 7 feet, and a height of 2 1/2 feet
36.
a trapezoid with bases 3/4 mile and 1 1/2 miles and a height of 1 mile
7
Area
UNIT THREE: Prealgebra in a Technical World
281
Area
37.
a circle with a radius of 16 feet 3 inches
38.
a circle with a diameter of 12 3/4 miles
39.
a circle with a diameter of 21 1/10 centimeters
40.
a circle with a radius of 3/4 inch
Applications
41.
Doc is making cookies for his Boy Scout troop. He doesn't do this often, so he wants to make
3
1
a lot and freeze the extras for another time. The recipe calls for 2 4 cups of flour and 1 2 cups
of sugar. If he wants to triple the recipe, how many cups of dry ingredients will he have all
together?
42.
Susan is barbecuing steak for a party. The package of meat that she thawed weighs 7 10 lbs.
If 10 people are coming to dinner, how much meat will there be per person?
43.
Emma is dieting. She records her weekly weight loss in pounds every Saturday: 1 2 , 0, − 2 ,
3
1
3
2, 1 4. What was her total weight loss for the 5 Saturdays?
3
44.
In one month, Ethan recorded the total gallons of gasoline he used, 52 10 gallons. At the
average price per gallon in his state of $2.77, he wanted to try to save money, so the next
4
month he only used 46 5 gallons of gas. To the nearest cent, how much money did he save
over the previous month if the price of gas didn't change?
45.
Mary is a dedicated runner. She runs 5 days a week, but her mileage varies, depending on
her schedule. She also keeps a record of her weekly miles. For one week in February, she
1
1
1
recorded 3 4 , 2 2 , 4, 1, 2 4. What was the average number of miles per day she ran that
week?
1
282
SECTION 3.8 Using Calculators
46.
1
3
1
Michael records his stock gains for the week. They were 108 , −18 4 , 0, −20, 12 2 . What was
the average gain/loss for the week?
Review and Extend
Use UPSand show work for credit.
47. Jenny has three arched windows in her dining room.
a. The top halves are semi-circles (1/2 of a circle) and the bottoms are
rectangular. If the rectangles measure 32 inches wide and 42 inches tall,
how many square feet is there in one window? Round your answer to the
nearest square foot.
1
3
2
4
b. Jerry is going to paint his dining room, which has 10 foot ceilings. The room is 12 feet by
1
15 2 feet, and he is going to paint the two shorter walls and one of the long walls. He will
subtract the area of the three windows to find out how many square feet he has to paint so
that he will know how much paint to buy. Can you find the area for him?
48.
Formulas that contain fractions can have very intricate math. One useful formula is the one that
calculates your monthly payment when you borrow an amount of money (P) at a given rate of
interest (r) for a number of months (t). For instance: If you financed $10,000 at 5% interest for 4
years = 48 months, your monthly payments would be found by using your calculator and the
following formula:
The formula for a monthly payment of M is
𝑟
12
M=
𝑟 −𝑡
1−(1+ )
12
And for this particular problem:
𝑃( )
𝑀=
0.05
10000 ( 12 )
0.05 −48
1 − (1 + 12 )
𝑀 is the monthly payment.
𝑃 = (amount financed) = $10,000
𝑟 = (annual interest rate) = 0.05
𝑡 = (the number of months) = 48
=
0.05
0.05 −48
10000 (
) ÷ [− (1 +
) ]
12
12
= $230.29
a. What would the monthly payment be if you financed $15,000 for 5 years (60 months)
at 6% interest? (Hint: take several steps and use your calculator for each step.)
b. How much interest would you pay? (Hint: Think of how much you paid over 60
months and compare it to the original amount financed.)