Download Section 1-1: Whole Numbers, Decimals, and the Place

Section 1-1: Whole Numbers, Decimals, and the Place-Value System
Learning Outcome 1
Identify the place value of each of the digits in the number 4,735: 4 thousands, 7 hundreds, 3
tens, 5 ones.
Learning Outcome 2
The number 451,375 is written in standard form. Write 2,853 in words: two thousand, eight
hundred fifty-three.
Write 2,042 in expanded notation.
2 x 1,000 + 0 x 100 + 4 x 10 + 2 x 1
Multiply each digit by its place value and add the products.
Learning Outcome 3
Write two inequalities comparing 4 and 7.
4<7
Using the less-than symbol, place the smaller number on the left.
7>4
Using the greater than symbol, place the larger number on the left.
Learning Outcome 4
In 6.28, 6 is in the ones place, 2 is in the tenths place, and 8 is in the hundredths place.
Learning Outcome 5
Write 57.035 in words: fifty-seven and thirty-five thousandths.
Learning Outcome 6
Write the numerator as a decimal with as many decimal places as the denominator has zeros.
Express
125
as a decimal: 0.0125. (The number has 4 decimal places because the
10, 000
denominator has 4 zeros.)
Learning Outcome 7
Which decimal is larger, 0.45 or 0.445? Both numbers have the same digit , 4, in the tenths place.
0.45 is larger because it has a 5 in the hundredths place, while 0.445 has a 4 in that place.
Learning Outcome 8
Round 4,734 to the nearest tens place: 3 is in the tens place; 4 is the digit to the right; 4 is less
than 5, so round down. The rounded value is 4,730.
Round 6.837 to the nearest hundredth: 3 is in the hundredths place; 7 is the digit to the right; 7 is
5 or more, so round up by raising the 3 to a 4 and dropping the 7. The rounded value is 6.84.
Learning Outcome 9
Round 3,682 to a number with one nonzero digit: 3 is the first nonzero digit on the left; 6 is the
digit to the right; 6 is 5 or more, so round up. The rounded value is 4,000.
Round 0.0683 to a number with one nonzero digit: 6 is the first nonzero digit on the left; 8 is 5 or
more, so round up by changing the 6 to a 7 and dropping the remaining digits on the right. The
rounded value is 0.07.
Section 1-2: Adding Whole Numbers and Decimal Numbers
Learning Outcome 1
4+2=2+4
6=6
Numbers may be added in any order.
(4 + 3) + 6 = 4 + (3 + 6)
7+6=4+9
13 = 13
Grouping does not change the sum.
0+5=5+0
5=5
Zero added to a number does not change the number.
4 + 5 = 9. Write 9 in the ones place.
2 + y = 6. Write 6 in the tens place.
7 + 7 = 14. Write 4 in the hundreds place and carry the 1 to next
column.
5 + 1 carried = 6. Write 6 in the thousands place.
5,724
+ 745
6,469
Learning Outcome 2
Add: 33.25 + 4.5 +0.123
a
33.25
4.5
+ 0.123
37.873
Align decimals in each addend in a vertical line.
Add each column starting at right. Keep decimal in answer
in same vertical line as in addends. Add the columns and carry
as needed.
Learning Outcome 3
Estimate the sum by rounding to the nearest hundred: 372 + 645.
372 rounds up to 400. 645 rounds down to 600. 400 + 600 = 1,000. Check by adding the
numbers a second time.
Exact sum: 1,017.
Add: 46.7 + 3,826 + 4,573
Calculator steps: 46[.]7 [+] 3826 [+] 4573 [=]
Equal key may also be labeled as [ENTER] or [EXE]
Sum = 8445.7.
Section 1-3: Subtracting Whole Numbers and Decimals
Learning Outcome 1
If 6 + 4 = 10, then 10 – 6 = 4 or 10 – 4 = 6.
Addition and subtraction are inverse operations.
6–3=3
Subtract the digits in the ones column.
To subtract 3 – 4 in the tens column, borrow 1 from the
5,436
4 in the hundreds column and make the 3 a 13; then 13 – 4 = 9.
– 243
Remember in the hundreds column the borrowed 1makes the 4
5,193
a 3, so 3 – 2 = 1.
Learning Outcome 2
Subtract: 68.029 – 6. 4276.
68.029
– 6.4276
61.6014
Align the decimals in a vertical line.
In the thousandths column, borrow 1 from the 9 to make
understood 0 in the ten-thousandth column a 10; then 10 – 6 =
4. In the ones column, borrow 1 from 8 to make the 0 in the
tenths column 10; then 10 – 4 = 6. Remember, in the ones
column 8 is now 7, 7 – 6 = 1.
Learning Outcome 3
Estimate the difference by rounding to the nearest thousand: 4,752 – 2,641.
5,000 – 3,000 = 2,000.
Exact difference: 4,752 – 2,641 = 2,111.
Check: 2,111 + 2,641 = 4,752. To check add the difference and the subtractand.
Section 1-4: Multiplying Whole Numbers and Decimals
Learning Outcome 1
3×5=5×3
15 = 15
The order of the numbers multiplied does not matter.
( 4 × 2) × 3 = 4 × (2 × 3 )
8 × 3=4 × 6
Grouping of numbers multiplied does not matter.
24 = 24
6 × 0=0 × 6
0=0
Multiplying by zero always gives zero as a product.
125
× 21
125
250
2, 625
Multiply multiplicand by ones digit of multiplier; put partial
product so ones digits are in line. Multiply by tens digit of
multiplier; place partial product so tens digits are in line.
Add the partial products
Learning Outcome 2
Multiply: 5.75 x 0.25
5.75
x 0.25
2875
1150
000
Multiply by each digit in the multiplier.
Since there are four decimal places total in multiplicand and
multiplier, make four decimal places in product.
1.4375
Learning Outcome 3
4(5 + 2) = 4(5) + 4(2)
4(7) = 20 + 8
28 = 28
The numbers in parentheses may be added (or subtracted) and
then multiplied by number outside parentheses. Or each
number in parentheses may be multiplied by number outside
parentheses and then added (or subtracted).
Learning Outcome 4
Estimate the product by rounding to one nonzero digit: 369 x 112. 400 x 100 = 40,000. Check by
multiplying a second time.
Exact product: 369 x 112 = 41,328.
Section 1-5: Dividing Whole Numbers and Decimals
Learning Outcome 1
Write 20 divided by 5 in four ways.
)
20 ‚ 5, 5 20,
20
, 20 5
5
Learning Outcome 2
21 R 8
22 470
)
44
30
22
8
47 ÷ 22 = 2. Place the 2 in the product over the 7 of
the minuend. 2 × 22 = 44; place the 44 under the 47 and
subtract. 47 – 44 = 3. Bring down the 0 from the minuend to
make the 3 a 30. 30 ÷ 22 = 1. Place the 1 in the
product over the 0 of the minuend. 1 × 22 = 22; place the 22
under the 30 and subtract. 30 – 22 = 8. 8 is the remainder.
Learning Outcome 3
Divide: 14.95 ÷ 2.3
6.5
23. 149.5
)
138
115
115
000
Move decimal in the divisor one place to the right. Move decimal
in dividend the same number of places, that is, 1 place. Place
decimal in quotient directly above decimal in dividend. Then
divide as with whole numbers.
Divide 30.5 ÷ 12 and round to tenths.
)
12
2.54 » 2.5
30.50
24
65
60
50
48
2
Since divisor is a whole number, no decimals are moved.
Place decimal in quotient directly over decimal in dividend.
Divide one place past tenths place. Since 4 is less than 5,
round 2.54 down to 2.5.
Divide 12.7 ÷ 6 to tenths and express remainder as a fraction.
2 .1
)
1
6
6 12.7
12
07
6
1
Since divisor is a whole number, no decimals are moved.
Place decimal in quotient directly over decimal in dividend.
Divide to tenths place. Write remainder, 1, over the divisor, 6, to
express remainder as a fraction.
Learning Outcome 4
Estimate 1,875 ÷ 36. Round divisor and dividend each to a number with one nonzero digit, then
divide to find first digit of quotient. Add zero(s).
50
40 2, 000
Exact quotient = 52Re.
Check: 52 × 36 = 1,872; 1,872 + 3 = 1,875
Divide: 455 ÷ 25
Basic or scientific calculator steps: 455 [÷] 25 [=]
Graphing calculator steps: 455 [÷] 25 [EXE]
Quotient = 18.2.
Learning Outcome 5
Find the average of 78, 85, and 96.
78 + 85 + 96 = 259
259 ÷ 3 = 86.33 or 86 (rounded)
Multiply quotient times divisor.
Add the remainder. The result should equal
original dividend.
Add the values.
Divide by the number of values.
Section 1-6: Exponents, Roots, and Powers of 10
Learning Outcome 1
34 = 3 × 3 × 3 × 3 = 81
81 = 8
60 = 1
3 is a factor 4 times.
a1 = a.
a0 = 1.
Learning Outcome 2
82= 64
64 = 8
8 × 8 = 64.
Since 82 is 64, 8 is the principal square root of 64.
Use a calculator to find 112, 153, 65, and 4.54.
112 11[x2] ⇒ 121 or 11[xy] 2 = ⇒ 121
153 15[x3] ⇒ 3375 or 15[xy] 3 = ⇒ 3375
65 6[xy] 5 [=]⇒ 7776
4.54 4.5[xy] 4 [=]⇒ 410.0625
Use a calculator to find 529 and
529 [
] 529 = ⇒ 23
20.25 [
] 20.25 = ⇒ 4.5
20.25 .
[ENTER] or [EXE] may
substitute for [=]
Learning Outcome 3
Multiply:
18 × 104= 180,000
25 × 100 = 2500
21.55 × 1,000 = 21,550
Divide:
4.65 ÷10 = 0.465
1650 ÷103= 1.650
45,000 ÷104= 4.5
Move decimal 4 places to the right.
Move decimal 2 places to the right.
Move decimal 3 places to the right.
Move decimal 1 place to the left.
Move decimal 3 places to the left.
Move decimal 4 places to the left.
Section 1-7: Order of Operations and Problem Solving
Learning Outcome 1
42+ 3(6 – 2) 2
42+ 3(4) 2
16 + 3(4) 2
16 + 12 2
16 + 6
22
Do subtracting in parentheses first.
Do exponential operations next.
Do multiplication.
Do division.
Do addition.
42 + 3(6 – 2) 2
Calculator: 4 [x2] [+] 3 [x] [(] 6 [–] 2 [)] [÷] 2 [=]