Download MAT018B

MAT018B
TOPICS: DECIMALS
RATIO & PROPORTION
4.1 Decimal Notation, Order, and Rounding:
I. Place value – There is no “oneths”! Look at the pattern of pairings below.
Hundreds tens ones tenths hundredths
Ex:
3
4
7.
1
9
Three Hundred FortySeven And Nineteen
Hundredths
II. Decimal- fractional conversion.
A. Decimal → fraction: count # of digits right of point and place over power of 10 with that # of
zeroes then try to reduce.
Ex: 0.17 =
Two two zeroes
B. Fraction → decimal: Generally divide up but for now use shortcut for powers of ten. You
want 1 downstairs so insert decimal points and then move them evenly to get 1 downstairs.
Then get rid of the 1.
III.
Ex:
=
=
= 0.003
three zeros
To decide which decimal number is larger, compare place-value by place-value from left to
right until tie is broken.
Ex: 0.136 < 0.137
tie breakers!
IV. Rounding, as before, involves underlining the place-value digit requested and looking right.
See 1.4.
Ex: round 1.378 to nearest hundredth.
≈1.38
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4.2
Addition and Subtraction
I. To add or subtract, line up vertically on decimal point and “carry” or “borrow” as necessary.
You may need to insert zeroes.
Ex: 2.781
-2. 0 8
0. 7 2
II. To solve equations (two ways).
A. Use “related sentence”.
Ex: w+67.8=100.23 w=100.23-67.8
B. Do “opposite” below on both sides a la algebra.
0 9 9 1
1 0 0. 23
Ex: w + 67.8
=
- 67.8
=
6 7. 8
w
=
3 2. 43
4.3
Multiplication:
I. The Rules!!
Step 1: Line numbers up vertically, flush right.
Step 2: Multiply ignoring points.
Step 3: Count # of digits right of point(s).
Step 4: Count in that number of digits from the right on your product to place point.
Ex: 9.8
.6
5.88
II. Powers of ten tricks:
{Don’t use this until you’ve gotten really sick of multiplying these out!}
A. Multiplying by 10,100,1000,…, count # of zeroes and move point right that many.
Ex: 0.3 × 100 = 30
two
B. Multiplying by 0.1, 0.01, 0.001…, count # of digits right of point and move it left that many
times.
Ex: 1.46 × 0.1 = 0.146
III. Converting.
A. $/¢: Try a proportion if confused. EX What is 34 cents in dollars?
Ex
=$0.34
B. 2.7 million means (2.7) (1,000,000).
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4.4
Division
I. The Rules,
Step 1 Set up with
Step 2 If need be, move point in outside # all the way right
Step 3 Move point in inside # that # right as well. (Add zeroes if necessary).
Step 4 Place point on top directly over inside point. Don’t move it again!
Step 5 Division as usual.
20.
Ex:
0.25 5.00
50
0
II. Powers of ten tricks:
These tricks are basically the opposites of those in 4.3.
Again, don’t use them until you’re sick of doing it the long way.
III. Solving equations… again: two ways.
A. Related sentence
Ex: 4.2x = 14 →x = 14 ÷4.2…
B. “Opposite” on both sides a la algebra
Ex:
IV. P.E.M.D.A.S. again!
4.5
Converting from fraction notation to decimal notation
I. The Rules:
A. The honest way again involves a proportion with 1 downstairs.
Ex: write as a decimal:
=
8x = 3, x = 3 ÷ 8
0.375
8 3
-24
60
-56
40
B. The trick: just divide up
II. Sometimes division finishes…and sometimes it repeats.
Ex:
0.33…
3 1.0
-9
10
-9
10
et cetera…
3
Put a bar over the fewest possible number of digits.
Ex: 0.3
(not 0.33)
III. Rounding uses the same rules as 4.1. (You may have to expand a repeating decimal to see
what’s going on).
Ex: round 1.17 to the nearest thousandth.
1.171717… ≈ 1.172
IV. When calculating with a mixture of fractions and decimals, try converting all fractions to
decimal because decimals are easier to work with. If anything repeats, convert all too fractional
form (putting decimals over one if necessary).
4.6
Estimating
Multiple choice answers can be found by process of elimination!
4.7
Applications and Problem Solving
I. As always… Please Let Frankie Sing! Or Pictures, Labels, Formulas, Substitutions! Also, if you
see a rate (“per”, “each”, etc.) try setting up a proportion!
5.1 Introduction to Ratios:
I. A “ratio” shows a relationship between two numbers.
Ex: 5 to 7 can be written as or as 5:7
II. “Simplified” fractions don’t have fractions or decimals within them and they are reduced.
Ex: 1.2 to 8.6;
×
=
=
5.2 Rates and Unit Price
I. A “rate” is a ratio that has different units on top, from those on the bottom
Ex:
(Often you divide up and attach the units with a slash. Ex 18.5 miles/hour)
II. “Unit price” is the amount of money for one. (That’s what unit means.)
The safest way to calculate it is again using a proportion!
Ex: What is unit price if 14 oz for 42¢?
=
=
= 3¢
III. Often a one is implied.
Ex “miles per hour” means “miles per one hour.
5.3 Proportions
I. Proportions are very powerful, and we have a deep natural understanding of them. Artists
consider them constantly.
II. A proportion is two ratios equal to each other.
Ex:
(This can be flipped in lots of ways and still be true.)
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III. To determine whether two pairs of numbers are proportional, the quick way is to cross multiply
and make sure the products are equal.
Ex: 3, 6, and 1, 2 →
2 ∙ 3 = 6 ∙ 1 yes
IV. Solving Equations
A. In algebra we “clear denominators.” The following trick is a version of that. We call it “cross
multiplying.” (It is not “cross canceling” which is something done when multiplying fractions!)
Ex: =
=
x = 12
Cross Multiply
Solve
5.4 Applications of Proportions
Step 1: Read! Often two numbers are related in one sentence.
Step 2: Write these numbers as a rate so include units.
Step 3: Set this equal to another “empty” fraction.
Step 4: Reread! Put x for unknown and number in final position using units consistently.
Step 5: Cross multiply and solve.
Ex Three gallons to cover 1275 ft2. How much covered with 7 gallons?
5.5 Geometric Applications
I. “Similar” triangles (and other shapes) have different sizes but the same “shape” because sides
are proportional. (All of trigonometry is based on this!)
Ex 3
5
is proportional to
15
4
9
12
Because… =
II. Many proportions can be formed using the sides of these similar triangles.
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