15
Section 1.3
DEFINITIONS & BASICS
Decimals
1) Like things – In addition and subtraction we must only deal with like things.
Example: If someone asks you
5 sheep + 2 sheep =
you would be able to tell them 7 sheep.
What if they asked you 5 sheep + 2 penguins =
We really can’t add them together, because they aren’t like things.
2) We do not need like things for multiplication and division.
3) Negative – The negative sign means “opposite direction.”
Example: −5.3 is just 5.3 in the opposite direction
−5.3
0
5.3
Example : − ళఴ is just ళఴ in the opposite direction.
Example: −7 – 5 = −12, because they are both headed in that direction
4) Decimal – Deci is a prefix meaning 10. Since every place value is either 10 times
larger or smaller than the place next to it, we call each place a decimal place.
5) Place Values – Every place on the left or right of the decimal holds a certain value
Arithmetic of Decimals, Positives and Negatives
LAWS & PROCESSES
Addition of Decimals
1. Line up decimals
2. Add in columns
3. Carry by 10’s
Section 1.3
16
EXAMPLE
Add. 3561.5 + 274.38
3561.5
+ 274.38
1. Line up decimals
3 5 6 1. 5
+ 2 7 4. 3 8
5. 8 8
2. Add in columns
1
3 5 6 1. 5
+ 2 7 4. 3 8
3 8 3 5. 8 8
3. Carry by 10’s. Carry the 1 and leave the 3.
Subtraction of Decimals
1.
2.
3.
4.
Biggest on top
Line up decimals; subtract in columns.
Borrow by 10’s
Strongest wins.
EXAMPLE
Subtract. 283.5 – 3,476.91
- 3476.91
283.5
- 3 4 7 6. 9 1
2 8 3. 5
3. 4 1
1.Biggest on top
2. Line up decimals; subtract in columns
3
- 3 4 17 6. 9 1
2 8 3. 5
3 1 9 3. 4 1
3. Borrow by 10’s. Carry the 1 and leave the 3.
3
- 3 4 17 6. 9 1
2 8 3. 5
- 3 1 9 3. 4 1
3. Biggest one wins.
Section 1.3
17
Multiplication of Decimals
Multiplication of Decimals
1. Multiply each place value
2. Carry by 10’s
3. Add
1. Add up zeros or decimals
4. Right size.
2. Negatives
EXAMPLES
Start:
7 5 31
3. Add the pieces
together.
29,742
× 538
237,936
892,260
+14,871,000
16,001,196
29,742
×
8
237,936
Next:
22 1
29,742
× 30
892,260
Last:
43 21
29,742
× 500
14,871,000
Section 1.3
18
Start:
Final example with decimals:
-7414.3
× 9.46
444858
2965720
+66728700
-70139278
3. Add the pieces together.
2 21
74143
×
6
444858
Next:
1 11
74143
× 40
2965720
Last:
3 132
4. Right size. Total number of
decimal places = 3. Answer is
negative.
74143
× 900
66728700
The only thing left is to count the number of decimal places. We
have one in the first number and two in the second. Final answer:
-70139.278
Division of Decimals
Division of Decimals
1. Move decimals
1. Set up. 2. Add zeros
2. Divide into first.
3. Multiply.
4. Subtract.
5. Drop down.
1. Remainder
6. Write answer.
2. Decimal
Section 1.3
19
EXAMPLES
5
8 429
Step 1. No decimals to set up. Go to Step 2.
Step 2.We know that 8 goes into 42 about 5
times.
Step 3. Multiply 5×8
5
8 429
Step 4.subtract.
-40
53
8 429
Step 5. Bring down the 9 to continue on.
Repeat steps 2-5
-40
29
53
8 429
Step 2: 8 goes into 29 about 3 times.
Step 3: Multiply 3×8
-40
29
-24
5
Step 4: subtract.
8 doesn’t go into 5 (remainder)
Which means that 429 ÷ 8 = 53 R 5
or in other words
429 ÷ 8 = 53 85
Example:
5875 ÷ 22
2
22 5875
Step 2: 22 goes into 58 about 2 times.
Step 3: Multiply 2×22 = 44
44
2
22 5875
Step 4: Subtract.
-44
147
27
22 5875
-44
147
154
Step 5: Bring down the next column
22 goes into 147 about ???? times.
Let’s estimate.
2 goes into 14 about 7 times – try that.
Multiply 22×7 = 154
Oops, a little too big
Section 1.3
20
26
22 5875
Since 7 was a little too big, try 6.
Multiply 6×22 = 132
-44
147
-132
155
267
22 5875
Subtract.
Bring down the next column.
22 goes into 155 about ????? times.
Estimate.
2 goes into 15 about 7 times. Try 7
-44
147
Multiply 22×7 = 154. It worked.
-132
155
Subtract.
-154
Remainder 1
1
5875 ÷ 22 = 267 R 1 or 267 221
An example resulting in a decimal:
4
Write as a decimal:
9
Step 1: Set it up. Write a few zeros, just to be
9 4.0000
safe.
Step 2: Divide into first.
.4
9 goes into 40 about 4 times.
9 4.0000
Step 3. Multiply 4×9 = 36
-36
4
Step 4. Subtract.
.44
9 4.0000
-36
40
-36
4
.444
9 4.0000
Repeating decimal
Step 5. Bring down the next column.
Repeat steps 2-4
Step 2: 9 goes into 40 about 4 times.
Step 3: Multiply 4×9 = 36
Step 4: Subtract.
Step 5. Bring down the next column.
Repeat steps 2-4
Step 2: 9 goes into 40 about 4 times.
Step 3: Multiply 4×9 = 36
-36
40
-36
Step 4: Subtract.
40
-36
This could go on forever!
4
4
Thus = .44444. . . which we simply write by .4
9
The bar signifies numbers or patterns that repeat.
Section 1.3
21
Two final examples:
358.4 ÷ -(.005)
.005 358.4
Step 1. Set it up and move the decimals
5 358400
7
5 358400
35
7
5 358400
-35
08
71
5 358400
-35
08
- 5
34
716
5 358400
-35
08
- 5
34
-30
40
7168
5 358400
-35
08
- 5
34
-30
40
-40
00
296 ÷ 3.1
3.1 296
31 2960.00
Step 2. Divide into first
Step 3. Multiply down
Step 4. Subtract
Step 5. Bring down
Repeat steps 2-5 as necessary
Step 2: Divide into first
Step 3: Multiply down
Step 4: Subtract
Step 5. Bring down
Repeat steps 2-5 as necessary
Step 2: Divide into first
Step 3: Multiply down
Step 4: Subtract
Step 5. Bring down
Repeat steps 2-5 as necessary
Step 2: Divide into first
Step 3: Multiply down
Step 4: Subtract
Step 5: Bring down
9
31 2960.00
279
9
31 2960.00
-279
170
95.
31 2960.00
-279
170
-155
150
95.4
31 2960.00
-279
170
-155
150
-124
26
95.48
31 2960.000
-279
170
-155
150
-124
260
- 248
120
Section 1.3
22
71680
5 358400
-35
08
- 5
34
-30
40
-40
00
-0
0
-71,680
Repeat steps 2-5 as necessary
Step 2: Divide into first
Step 3: Multiply down
Step 4: Subtract
95.483
31 2960.000
Step 6: Write answer
95.483 . . .
One negative in the original problem gives
a negative answer.
-279
170
-155
150
-124
260
- 248
120
-93
27
The decimal obviously keeps going.
Round after a couple of decimal places.
COMMON MISTAKES
Two negatives make a positive
- True in Multiplication and Division – Since a negative sign simply means
opposite direction, when we switch direction twice, we are headed back the way we
started.
Example: -(-5) = 5
Example: -(-2)(-1)(-3)(-5) = - - - - -30 = -30
Example: -(-40 ÷ -8) = -(- -5) = -5
- False in Addition and Subtraction – With addition and subtraction negatives
and positives work against each other in a sort of tug ‘o war. Whichever one is stronger
will win.
Example: Debt is negative and income is positive. If there is more debt than
income, then the net result is debt. If we are $77 in debt and get income of $66
then we have a net debt of $11
-77 + 66 = -11
On the other hand if we have $77 dollars of income and $66 of debt, then the net
is a positive $11
77 – 66 = 11
Section 1.3
23
Example: Falling is negative and rising is positive. An airplane rises 307 feet and
then falls 23 feet, then the result is a rise of 284 feet:
307 – 23 = 284
If, however, the airplane falls 307 feet and then rises 23 feet, then the result is a
fall of 284 feet:
-307 + 23 = -284
Other examples: Discount is negative and markup or sales tax is positive.
Warmer is positive and colder is negative. Whichever is greater will give you the
sign of the net result.
1) Percent: Percent can be broken up into two words: “per” and “cent” meaning per hundred,
or in other words, hundredths.
7
31
53
= .07 = 7%
= .31 = 31%
= .53 = 53%
100
100
100
Notice the shortcut from decimal to percents: move the decimal right two places.
Example:
LAWS & PROCESSES
Converting Percents
Percents
1. If fraction, solve for decimals.
for decimal to %
2. Move decimal 2 places. 1.2. Right
Left for % to decimal
3. “OF” means times.
EXAMPLES
.25=
25%
Convert .25 to a percent
Move the decimal two places to the right because
we are turning this into a percent
.25=25%
Section 1.3
24
What is
5 ÷ 32 = .15625
.15625=15.625%
ହ
as a percent?
ଷଶ
Turn the fraction into a decimal by dividing
Move the decimal two places to the right because
we are turning this into a percent
5
= 15.652%
32
124%=1.24
Convert 124% to decimals
Move the decimal two places to the left because
we are turning this into a decimal
124%=1.24
Solving “Of” with Percents
The most important thing that you should know about percents is that they never stand
alone. If I were to call out that I owned 35%, the immediate response is, “35% of what?”
Percents always are a percent of something. For example, sales tax is about 6% or 7% of
your purchase. Since this is so common, we need to know how to calculate this.
If you buy $25 worth of food and the sales tax is 7%, then the actual tax is 7% of $25.
.07×$25 = $1.75
In math terms
the word “of”
means multiply.
EXAMPLES
25%=.25
. 25 × 64 = 16
What is 25% of 64?
Turn the percent into a decimal
Multiply the two numbers together
25% of 64 is 16
What is 13% of $25?
13%=.13
. 13 × 25 = 3.25
30%=.30
. 30 × 90 = 27
Turn the percent into a decimal
Multiply the two numbers together
13% of $25 is $3.25
What is 30% of 90 feet?
Turn the percent into a decimal
Multiply the two numbers together
30% of 90 feet is 27 feet
Section 1.3