Download Ratios

5.1
Ratios
5.1
OBJECTIVES
1. Write the ratio of two numbers in simplest form
2. Write the ratio of two quantities in simplest form
In Chapter 2, you saw two meanings for a fraction:
1. A fraction can name a certain number of parts of a whole.
3
names 3 parts of a whole
5
that has been divided into 5 equal parts.
3
2. A fraction can indicate division. can be thought of as 3 5.
5
We now want to turn to a third meaning for a fraction:
3. A fraction can be a ratio. A ratio is a means of comparing two numbers or quantities.
Example 1
NOTE Another way of writing
the ratio of 3 to 5 is 3:5. We
have chosen to use only the
fraction notation for a ratio in
this textbook.
Writing a Ratio as a Fraction
Write the ratio 3 to 5 as a fraction.
3
3
To compare 3 to 5, we write the ratio of 3 to 5 as . So also means “the ratio of 3
5
5
to 5.”
CHECK YOURSELF 1
Write the ratio of 7 to 12 as a fraction.
Example 2 illustrates the use of a ratio in comparing like quantities, which means we’re
comparing inches to inches, cm to cm, apples to apples, etc.
Example 2
Applying the Concept of Ratio
The width of a rectangle is 7 cm and its length is 19 cm. Write the ratio of its width to its
length as a fraction.
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7 cm
7
19 cm
19
NOTE A ratio fraction can be
greater than 1.
NOTE In this case the ratio is
never written as a mixed
number. It is left as an improper
fraction.
We are comparing centimeters to centimeters,
so the units “cancel.”
The ratio of its length to its width is
19 cm
19
7 cm
7
CHECK YOURSELF 2
A basketball team wins 17 of its 29 games in a season.
(a) Write the ratio of wins to games played.
(b) Write the ratio of wins to losses.
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414
CHAPTER 5
RATIOS AND PROPORTIONS
Because a ratio is a fraction, we can reduce it to simplest form. Consider Example 3.
Example 3
NOTE When simplifying a
Writing a Ratio in Simplest Form
fraction, you are actually
multiplying by one.
Write the ratio of 20 to 30 in lowest terms.
1
20 10
30
1
10
20
2
30
3
Divide the numerator and denominator
by the common factor of 10.
CHECK YOURSELF 3
Write the ratio of 24 to 32 in lowest terms.
Our next example relates to an application of ratios.
Example 4
Simplifying the Ratio of Two Dimensions
A common size for a movie screen is 32 ft by 18 ft. Write this as a ratio in simplest form.
32 ft
32
16
18 ft
18
9
CHECK YOURSELF 4
A common computer display mode is 640 pixels (picture elements) by 480 pixels.
Write this as a ratio in simplest form.
Some ratios include fractions or decimals, as in the next two examples.
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20 10
is the same as
30 10
RATIOS
SECTION 5.1
415
Example 5
Simplifying a Ratio Involving a Fraction
1
Loren sank a 22 foot putt and Carrie sank a 30 foot putt. Express the ratio of the two dis2
tances as a ratio of whole numbers.
1
45
45
2
2
2
30
30
30
1
22
Because we are dividing a fraction by a fraction, we invert and multiply.
45
30
45
1
3
2
1
2
30
4
1
The ratio 22 to 30 is equivalent to the ratio 3 to 4.
2
CHECK YOURSELF 5
1
1
Rita jogged 3 miles this morning, and Yi jogged 4 miles. Express the ratio of the
2
4
two distances as a ratio of whole numbers.
The next example simplifies a ratio involving decimals.
Example 6
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Simplifying a Ratio Involving Decimals
The diameter of a 20-oz bottle is 2.8 inches (in.), and the diameter of a 2-liter bottle is
5.25 in. Express the ratio of the two diameters as a ratio of whole numbers.
2.8
2.8 100
280
5.25
5.25 100
525
280
8
525
15
The ratio of the diameters 2.8 to 5.25 is equivalent to the ratio 8 to 15.
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CHAPTER 5
RATIOS AND PROPORTIONS
CHECK YOURSELF 6
The width of a standard newspaper column is 2.625 in., and the length of a standard
column is 19.5 in. Express the ratio of the two measurements as a ratio of whole
numbers.
In our final example in this section, we will see that sometimes, to find a ratio, we must
rewrite two denominate numbers so that they have the same dimensions.
Example 7
Rewriting Denominate Numbers to Find a Ratio
2 hours 60 min 120 min
1
1 hour
1
2 hours
120 minutes
8
75 minutes
75 minutes
5
CHECK YOURSELF 7
Find the ratio of whole numbers that is equivalent to the ratio of 16 feet to 10 yards.
CHECK YOURSELF ANSWERS
7
12
7
6.
52
1.
2. (a)
7.
17
17
; (b)
(the team lost 12 games)
29
12
3.
3
4
4.
4
3
5.
14
17
8
15
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NOTE
Joe took two full hours to complete his final, but Jaymie finished hers in 75 minutes. Find
the ratio of the two times.
To find a ratio, both numbers must have the same units. If we first convert the two hours
to 120 minutes, both units are minutes.
Name
5.1
Exercises
Section
Date
Write each of the following ratios in simplest form.
1. The ratio of 9 to 13
ANSWERS
2. The ratio of 5 to 4
1.
3. The ratio of 9 to 4
4. The ratio of 5 to 12
5. The ratio of 10 to 15
6. The ratio of 16 to 12
2.
3.
4.
1
2
7. The ratio of 3 to 14
3
5
8. The ratio of 5 to 2
1
10
5.
6.
9. The ratio of 10.5 to 2.7
10. The ratio of 2.2 to 0.6
7.
8.
11. The ratio of 12 miles (mi) to 18 mi
12. The ratio of 100 centimeters (cm)
to 90 cm
9.
10.
13. The ratio of 40 ft to 65 ft
14. The ratio of 12 oz to 18 oz
15. The ratio of $48 to $42
16. The ratio of 20 ft to 24 ft
11.
12.
13.
17. The ratio of 75 seconds (s) to
18. The ratio of 7 oz to 3 lb
14.
3 minutes (min)
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15.
19. The ratio of 4 nickels to 5 dimes
20. The ratio of 8 in. to 3 ft
21. The ratio of 2 days to 10 h
22. The ratio of 4 ft to 4 yd
23. The ratio of 5 gallons (gal) to
24. The ratio of 7 dimes to 3 quarters
12 quarts (qt)
16.
17.
18.
19.
20.
21.
22.
23.
24.
417
ANSWERS
Solve the following applications.
25.
25. Class make-up ratio. An algebra class has 7 men and 13 women. Write the ratio of
26.
men to women. Write the ratio of women to men.
27.
26. Football ratio. A football team wins 9 of its 16 games with no ties. Write the ratio
of wins to games played. Write the ratio of wins to losses.
28.
29.
30.
31.
32.
33.
27. Election ratio. In a school election 4500 yes votes were cast, and 3000 no votes
were cast. Write the ratio of yes to no votes.
28. Basketball ratio. A basketball player made 42 of the 70 shots taken in a tournament.
Write the ratio of shots made to shots taken.
1
3
4
8
Express the ratio of the two distances as a ratio of whole numbers.
29. Carla walked 2 miles (mi) this afternoon and Mario walked 5 mi this afternoon.
1
2
of the capacities as a ratio of whole numbers.
3
4
30. One car has an 11 gallon tank and another has a 17 gallon tank. Express the ratio
3
2
3
4
Express the ratio of the capacities as a ratio of whole numbers.
32. The price of an antibiotic in one drugstore is $12.50 although the price of the same
antibiotic in another drugstore is $8.75. Write the ratio of the prices as a ratio of
whole numbers.
33. The width of a notebook is 3.5 inches (in.) and the length is 6.75 in. Write the ratio of
length to width as a ratio of whole numbers.
418
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31. One refrigerator holds 2 cubic feet of food and another holds 5 cubic feet of food.
ANSWERS
34. Marc took 3 hours (h) to mow a lawn although Angelina took 150 minutes (min) to
mow the same lawn a week earlier. Write the ratio of Marc’s time to Angela’s time as
a ratio of whole numbers.
34.
35.
36.
37.
35. Employment ratio. A company employs 24 women and 18 men. Write the ratio of
men to women employed by the company.
36. Measurement ratio. If a room is 30 ft long by 6 yards wide, write the ratio of the
length to the width of the room.
37. (a) Buy a 1.69-oz (medium size) bag of M&Ms. Determine the ratio of the number
of M&Ms of each color (yellow, red, blue, orange, brown, and green) to the total
number of M&Ms in the bag.
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(b) Compare your ratios to those your classmates obtain.
(c) Use the information from parts a and b to determine a ratio for all the different
colors in a bag of M&Ms.
(d) E-mail the manufacturer of M&Ms (Mars, Inc.) at www.m-ms.com and see if
they use fixed ratios to determine the distribution of the colors in a bag. If they
do, compare these ratios to yours.
419
ANSWERS
38. Two pencils are shown. Write the ratio of the length of the smaller pencil to the larger
38.
pencil.
Answers
9
13
5
17.
12
32
31.
69
3.
9
4
2
5
27
33.
14
19.
5.
2
3
24
5
3
35.
4
21.
7.
1
4
9.
23.
5
3
35
9
2
3
7 13
;
25.
13 7
11.
8
13
3
27.
2
13.
8
7
18
29.
43
15.
37.
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1.
420