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Skills Worksheet
Science Skills
Ratios and Proportions
HOW RATIOS COMPARE NUMBERS
If you buy a bag of multicolored candies, you may notice that there are not equal
numbers of each color. For example, an average bag may contain twice as many
red candies as green ones. A comparison between two numbers, such as the number of red and green candies, is called a ratio.
Ratios can be expressed in many different ways. For example, you can express
the ratio of red to green candies in a bag of candy as follows:
• twice as many
• 2 to 1
• 2:1
• 2/1
•
2
1
WHAT RATIOS MEAN
Although these expressions look different, they each mean the same thing. Each
tells you that for every green candy, there are two red candies. So, if there are
three green candies, there are six red ones, and if there are five green candies,
there are ten red ones.
RATIOS DO NOT GIVE TOTAL NUMBERS
Note that a ratio does not tell you the total number of objects. If you have a ratio
of 3 : 1 brown to green candies, there may be a total of four candies, or eight, or
twelve, and so on. All you know for sure is that for each green candy, there are
three brown ones.
HOW TO SEE IF RATIOS ARE EQUAL
It is often useful to compare ratios and see if they are equal. If two ratios are
equal, they are proportional, and one can be reduced to the other. For example,
the ratio 6 : 3 is proportional to the ratio 2 : 1 because 6/3 reduces to 2/1. To verify
that two ratios are equal, you can write the ratios in fraction form and cross-multiply them:
6 2
3 1
In this case, 6 1 3 2, so the ratios are equal.
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Science Skills continued
KEEPING RATIOS PROPORTIONAL
It is often important to keep ratios in proportion. For example, when you are
cooking, if you double one ingredient in a recipe, you must double all of the other
ingredients as well so that the proportions between them remain the same. If you
are making chocolate-chip cookies, for instance, and you double the quantity of
dough, you need to double the quantity of chocolate chips also, or your cookies
will not have enough chips.
Problem
The ratio of blue to yellow candies in a bag is 3 : 2. If the bag contains 28
yellow candies, how many blue candies are there?
Solution
Step 1: Write the proportion as two equal fractions, using x for the
unknown number. Be sure to write each ratio in the same order. In this case,
blue candies is in the numerator of both fractions.
x blue candies
3 blue candies
2 yellow candies 28 yellow candies
Step 2: Cross-multiply the two fractions to set up an equation. Multiply
the numerator of each side by the denominator of the other side. The fractions
are equal, so these two quantities must also be equal.
3 28 2 x
Step 3: Isolate the unknown variable, x. In this case, divide each side by 2.
3 28 2 x
x
2
2
Step 4: Solve the equation. Because x is the value for the unknown part of the
proportion from step 1, solving for x will complete the proportion.
x
3 28
42 blue candies
2
Step 5: Check your answer. Rewrite the proportion, substituting the value for
x. When you cross-multiply, the two values should be equal.
3 blue candies
42 blue candies
2 yellow candies 28 yellow candies
3 28 2 42
84 84
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Science Skills continued
Practice
Work out the practice problems on a separate sheet of paper, and write your
answers in the spaces provided.
1. A recipe for bread says that the ratio of flour to water should be 3 : 2. If you
use 9 cups of flour, how much water will you need?
2. If the ratio of male to female teachers in your school is 3 to 5, and there are
35 female teachers, how many male teachers are there?
3. Solve for x in the proportion 100:30 2000: x.
4. An athlete rides his bike 8 miles for every 3 miles that he runs. If the athlete
runs 4.5 miles, how far does he ride his bike?
5. If the ratio of beakers to test tubes in a lab is 3:8, and there are 64 test tubes,
how many beakers are there?
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TEACHER RESOURCE PAGE
4. m D V
Science Skills
m
MAKING AND INTERPRETING BAR
GRAPHS AND PIE CHARTS
1. August
2. drama
m
Students’ bar graphs should have each
month listed on the horizontal axis, and a
scale from 0 to 5 on the vertical axis. The
horizontal and vertical axes should be
labeled “Months” and “Number of Movies
Seen,” respectively. The heights for each
bar should be as follows: Jan: 3, Feb: 2,
Mar: 1, Apr: 1, May: 3, June: 4, July: 5, Aug:
4, Sep: 2, Oct: 2, Nov: 0, Dec: 4.
2. Making a Pie Graph
Students’ pie graphs should have five
pieces labeled action, drama, comedy,
romance, and documentary. The relative
sizes of the pieces should be approximately
as follows: action 42%; drama 25%; comedy
22%; romance 10%. (Total does not add up
to 100% due to rounding.) The exact size of
each piece is not as significant as the relative sizes between pieces, for example
comedy should be slightly smaller than
drama, action should be the largest, etc.
RATIOS AND PROPORTIONS
4. 12 miles
5. 24 beakers
REARRANGING ALGEBRAIC
EQUATIONS
3. w V/(l h)
4. t E/(cm)
Math Skills
DENSITY
1. m D V
10 g
3.03 cm
(1.40 cm )
14
3
3
4.24 1014 g
m 4.24 1011 kg
m
6. V D
9.56 g
V
5.49 cm3
1.74 g/cm3
m
7. V D
432 g
V
160 cm3
2.7 g/cm3
m
8. V D
1.5 g
V
0.43 cm3
3.51 g/cm3
m
9. V D
286 kg
103 g
V
kg
19.32 g/cm3
1.48 104 cm3
m
10. V D
2.77 kg
103 g
V
kg
0.70 g/cm3
4.0 103 cm3
m
11. D V
1.58 1012 g
D
3.16 1015 g/cm3
500.0 cm3
m
12. D V
525 g
D
0.70 g/cm3
750 cm3
The liquid is most likely to be gasoline.
13. m D V
1.22 g
m
(1230 cm3)
cm3
1.50 103 g
m 1.50 kg
m
14. V D
2.000 kg
103 g
V
1 kg
21.45 g/cm3
93.24 cm3
1. Making a Bar Graph
1. l A/w
2. t d/v
3
3
5. m D V
3. 25%
1. 6 cups of water
2. 21 male teachers
3. x 600
g
(34.17 cm ) 766.1 g
22.42
cm 3.97 g
m
(575 cm3) 2280 g
cm3
m 2.28 kg
2. m D V
0.250 g
m
(87.3 cm3) 21.8 g
cm3
3. m D V
22.5 g
(43.2 cm3) 972 g
m
cm3
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