Decimals, Ratio, Proportion and Percent

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Chapter 7

• Decimals, Ratio, Proportion and Percent

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7.1 Decimals

• Decimals are used to represent fractions in our
  base ten place-value notation.

1000
100
10
1
1/10
1/100
1/1000
3
4
5
7
9
6
8
thousands
hundreds
tens
ones
tenths
hundredths
thousandths
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• Write 3457.968 in expanded form

Write 978.314 in expanded form
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• Theorem Let be a fraction in simplest form.
• Then has a terminating decimal
  representation
• if and only if b contains only 2s and/or 5s in
  its
• prime factorization.

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Ordering Decimals

• Terminating decimals can be compared using a
  hundreds square, using a number line, by
  comparing them in their fraction form, or by
  comparing place values one at a time from left to
  right just as we compare whole numbers.

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Multiplying or Dividing by Powers of 10

• Theorem Let n be any decimal number and m
  represent any nonzero whole number.
• Multiplying a number n by 10m is equivalent to
  forming a new number by moving the decimal point
  of n to the right m places.
• Dividing a number n by 10m is equivalent to
  forming a new number by moving the decimal point
  of n to the left m places.

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7.2 Operations with Decimals

• Addition and Subtraction requires the numbers to
  be aligned at the decimal point, add or subtract
  the numbers as if they were whole numbers, and
  insert a decimal point directly below the numbers
  being added or subtracted.

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Multiplication of Decimals

• Perform multiplication as if the decimal point
  were not there. Insert a decimal point in the
  answer as follows The number of digits to the
  right of the decimal point in the answer is the
  sum of the number of digits to the right of the
  decimal points in the numbers being multiplied.

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Division of Decimals

• Multiply both the divisor and dividend by the
  power of ten necessary to result in the divisor
  being a whole number.

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Repeating Decimals
Theorem Let be a fraction in simplest
form. Then has a repeating decimal
representation that does not terminate if and
only if b has a prime factor other than 2 or 5.
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7.3 Ratios and Proportion

• Definition A ratio is an ordered pair of
  numbers, written ab, with b not equal to 0.
• Definition Equality of Ratios
• if and only if

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Proportions

• Definition A proportion is a statement that two
  given ratios are equal.

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Rates

• Ratios involving different units are called
  rates.
• Examples miles per hour, miles per gallon,
  cents per ounce, dollars per pound, etc.

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Example 7.21 Adams School orders 3 cartons of
chocolate milk for every 7 students. If there
are 581 students in the school, how many cartons
of chocolate milk should be ordered?
n cartons to be ordered
Step 1 Define your variable.

• Step 2 Set up a proportion.

Step 3 Cross multiply and solve.
The school needs to order 249 cartons of
chocolate milk.
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1. Example 7.24 In a scale drawing, 0.5 centimeter
   represents 35 miles. How many miles will 4
   centimeters represent?

n of miles
Step 1 Define your variable.
Step 2 Set up a proportion.
Step 3 Cross multiply and solve.
Four centimeters represents 280 miles.
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7.4 Percent
Percent means per hundred
Fractions

• Percents

Repeating Decimals
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Conversions
Fraction Decimal Percent



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Solving Percent Problems

• Grid Approach

                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
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Example 7.29a A car was purchased for 13000
with a 20 down payment. How much was the down
payment?

• Shade 20 out of the 100 squares (20)

                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
100 squares 13000, so 1 square 130.
20 squares represent the down payment of
2600. 20 X 130 2600
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Solving Percent Problems

• Proportion Approach
• Since percents can be written as ratios, percent
  problems can be solved using proportions.

Ex 7.30a
The down payment was 2600.
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Solving Percent Problems

• Equation Approach

Translation of Problem Equation