Chapter 6
Notes 6-1 Ratios and Unit Rates
Ratio – a fraction – a comparison of two numbers with division
10 to 15
10:15
10
15
Rate – a ratio
Unit Rate – a ratio with a denominator of 1, example price each
To find a unit rate – Divide the numerator by the denominator
example 10 red pens for $2.00 = 2 ÷ 10 = .2 = $0.20
This is 20 cents each.
To change minutes to hours
multiply by a fraction that is equal to 1. 60 min. = 1 hour
1.5 gallons
per minute
.
60 minutes = 90 gallons
1 hour
per hour
(Use the charts in from the book
for pints, cups, miles, yards, etc.)
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Notes 6-2
Proportions
Proportion – equality of two ratios
To do cross products –
multiply the numerator of the 1st fraction with the denominator of the 2nd fraction
multiply the numerator of the 2nd fraction with the denominator of the 1st fraction
These are equal.
example 3 =
7.
If there is an unknown, solve for the variable.
9
x
3x = 7 • 9
3x = 63
3
3
x = 21
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Notes 6-3
Similar Figures and Scale Drawings
Similar figures ~ have the same shape, but not always the same size.
They have corresponding sides, sides are PROPORTIONAL.
They have corresponding angles, angles are EQUAL.
Δabc ~ Δdef
The angles of the small triangle equal the angles of the large one, ⁄ a = ⁄ d, ⁄ b = ⁄ e, ⁄ c = ⁄ f .
the small bottom side is proportional to the large bottom side, ac ~ df
the small right side is proportional to the large right side, bc ~ ef
and the small left side is proportional to the large left side. ab ~ de
We can use the information on similar figures to find a missing side, or an
indirect measurement – a measurement made without a ruler or measuring tape.
ex. – Find the height of the flag pole using the shadows. Set up the proportion. Use cross products. Solve.
Notes 6-4
Fractions, Decimals, and Percents
Writing a Percent as a Fraction – write the percent as a fraction with the denominator of
100. Then simplify. ex. 50% = 50/100 = ½
Writing a Percent as a Decimal – divide by 100, by moving the decimal point two places
to the left.
ex. 52.% = .52
ex. 2.% = .02
Writing a Decimal as a Percent – multiply by 100, by moving the decimal point two
places to the right and adding the percent sign. ex. .52 = 52%
ex. .02 = 2%
Writing a Decimal as a Fraction – write the decimal as a fraction with the denominator
of one and add zeros for each decimal place. Simplify in needed.
Writing a Fraction as a Decimal – Divide the numerator by the denominator.
Writing a Fraction as a Percent – Change the fraction to a decimal, then move the
decimal two places to the left, or create a proportion with the denominator of the second
fraction equal to 100. Solve by equivalent fraction rules, or use cross products.
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Notes 6-5
Proportions and Percents
Solve percent problems by using proportions – part is to whole, as percent is to 100.
Solve by using rules of equivalent fractions, or cross products.
Set up as “is” over “of”
42 “is” a percent “of” x.
The “part” is on top, and the “whole” is below.
Percent “is” out “of” 100
Notes 6-6
Percents and Equations
Solve by translating the “words” into mathematical symbols.
% means change it to a decimal
of means multiply
is means equal
∏ means “unknown” use a variable and solve for x.
ex. 80% of 20 is ?
ex. ?% of 40 is 30
15% of ? is 9
.80 • 20 = x
x% • 40 = 30
.15 • x = 9
multiply to solve
divide both sides by 40
divide both sides by .15
then change the decimal to a percent
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Notes 6-7
Percent of Change
Increase means goes up
Decrease means goes down
Find the Percent of Change
find the difference between the number you START with , and the END number,
then divide the amount of change by the amount you STARTed with.
ex. From 180 to 108, Find the Percent of Change.
180 – 108 = 72 This is the amount of change
72 ÷ 180 = 0.4 This is the answer as a decimal. Change it to a Percent.
0.4 • 100 = 40% This is the FINAL answer. 40% decrease
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Notes 6-8
Markup and Discount
A Markup is an increased amount. ADDED to the cost of the item.
The Percent of Markup is the percent of increase.
To find the amount of Markup – multiply the percent of markup, by the cost of the item.
ex. Jacket costs $20 with 20% markup. $20 • .20 = $4.00
To find the selling price, add the amount of Markup to the cost of the item.
$20 jacket + $4 = $24.00 selling price of jacket
A Discount is a decreased amount. SUBTRACTED from the price of the item.
The Percent of Discount is the percent of decrease.
To find the amount of discount –
multiply the selling price, by the percent of discount in decimal form.
ex. $140 guitar with a 10% discount
$140 • .10 = $14
To find the sale price – subtract the amount of discount from the price.
$140 guitar – $14 discount = $126 sale price
(Sometimes you can multiply the remaining percent by the price, and skip the last step.)
Example 40% off on a $200 ipod.
or, because 40% + 60% = 100%,
.40 • $200 = $80
60% • $200 = $120
$200 – $80 = $120
Commission – the amount of money a sales person earns making a sale
You can find this by using the equation:
Amount of Sale • percent of commission = money earned (This is the Commission.)
example – A sales person made $30 selling merchandise. Her commission rate was 5%.
What was the amount of the sale? x • .05 = 30
Divide both sides by .05.
x = $600