RATIO AND PROPORTION



RATIO AND PROPORTION

A ratio expresses a relationship between two numbers. The ratio of the numbers a and b may be symbolized by a:b, or as a fraction, [pic]. Both express the ratio a to b.

The equality of two ratios is called a proportion. Thus, if a compares to b in the same ratio as c compares to d we have a proportion that may be expressed as

a:b = c:d or [pic] = [pic]

Either form may be read, “a is to b as c is to d.”

When any one of the four numbers, a, b, c, or d is unknown and the other three are known, we may find the unknown by solving the proportion.

Assign a variable to represent the unknown number and express the proportion

as the equality of two ratio fractions.

Solve the resulting equation for the variable.

For example, [pic] = [pic][pic]. Since this is an equation we may solve it for x by

“doing the same thing to both sides.” This is a good time for a little digression

on solving equations. Remember a good symbol of an equation is a balanced

scale To keep it balanced, if you add something to one

side you must add the same amount to the other side. If you remove (subtract) something

from one side you must remove (subtract) the same amount from the other side.

2x+3 = 15

Take 3 from the left side and 3 from the right side:

2x = 12

Divide each side by 2:

x = 6

Let’s apply this to our simple proportion, [pic] = [pic] (A)

First multiply both sides by 15:

15 . [pic] = [pic] .[pic]15

and simplify [pic][pic][pic] = x

x = 5[pic]

A proportion can always be solved in this manner. Two more examples:

(B)[pic][pic] = [pic] (C) [pic] = [pic]

35 . [pic] = [pic] . 35 x . [pic] = [pic] . x

[pic] = x [pic] x [pic] = 11 . [pic]

19[pic] = x x = [pic]

x = 26 [pic]

Any complications from the above procedures can be eliminated by using cross multiplication; that is, by multiplying the numerator of one fraction by the denominator of the other. Let’s try it with two fractions we know are equal:

[pic] = [pic] means 5x2 = 1x10

10=10 So, when two fractions are equal, it follows that their “cross products” are equal. Another example:

[pic] = [pic] The cross products are 3x20 and 2x30. Both equal 60.

We’ll apply this method to our earlier examples.

(A) [pic] = [pic] (B) [pic] = [pic] (C) [pic]= [pic]

8x = 45 9x = 5 (35) 5x = 132

x = 5[pic] 9x = 175 x = 26 [pic]

x = 19[pic]

Notice especially how (C) which has x, the variable, in the denominator is now no more complicated than (A) or (B).

PRACTICE

Solve for x.

1. [pic] = [pic] 2. [pic] = [pic] 3. [pic] = [pic]

4. [pic] = [pic] 5. [pic] = [pic]

6. If the ratio of a to b is 3 to 1, what is the value of a if b = 24?

7. If three oranges cost 90 cents, how much will 12 oranges cost?

8. If one inch equals 2[pic]centimeters, how many inches equal 100 centimeters?

9. If 90 feet of wire weighs 18 pounds, what will 110 feet of the same wire weigh?

10. If 100 capsules contain [pic] of a grain of medication, how many capsules contain [pic] of a grain of medication?

11. If a recipe that calls for 2 cups of sugar makes 36 cookies, how many cookies will 3 cups of sugar make?

You may write this proportion, [pic] = [pic]

“Two cups of sugar is to 36 cookies as 3 cups of sugar is to x cookies.”

You may also write [pic] = [pic]

You can see that [pic] = [pic] which gives 2x = 3 36

is the same as [pic] = [pic] which also gives 2x = 3 36.

That should let us see that either way gives the same result. Whether we have [pic] = [pic] or [pic][pic] = [pic], since we are dealing with a direct relationship, a direct proportion, we will get the desired result. It follows that, whether we keep our cups together in one ratio and our cookies together in the other ratio or keep the cups with the number of cookies they produce in each ratio we get the same result in a direct proportion.

[pic] = [pic] or [pic] = [pic]

2x = 108 2x = 108

x = 54 x = 54

Three cups of sugar will produce 54 cookies! All this should seem very logical. When the amount of sugar increases the number of cookies increases.

There are situations when one quantity increases the other decreases. In these cases we are dealing with inverse proportions. For example, consider this problem:

When Lucy bikes to school at 10 miles per hour it takes her 15 minutes. How long will it take if she walks to school at 4 miles per hour?

Clearly, the faster she goes the shorter the time it takes to complete the trip. If we set up a ratio between the speeds, [pic], the time that corresponds to the 10 mph rate will be less than the time for the 4 mph rate. [pic] We cannot put the time that corresponds to the 10 mph in the numerator

[pic] = _______ because then the denominator has to be smaller than the numerator to equal [pic]. Why? This would lead to the illogical result that the time for the slower speed,

would be less than the time for the faster speed! Hence, when we set up the inverse proportion we write

[pic] = [pic]

4x = 150

x = 37[pic] minutes which is sensible. If it takes 15 minutes when you’re traveling at 10 miles per hour, it will take longer (37[pic] minutes) when you slow down to 4 miles per hour.

There are several common examples of inverse relationship:

Travel faster; take less time.

Work faster, take less time.

Increase pressure on a cylinder of gas; decrease the volume.

CAUTION

Remember how casual we could be with direct proportion?

(A) [pic] = [pic] or (B) [pic] = [pic]

x = 54, no matter whether you mixed your cups and cookies in a given ratio, (A), or made a separate ratio of cups and one of cookies, (B).

For inverse proportion, we must not mix our quantities. In our example above about riding and walking to school, we must make one ratio of rates (miles per hour) and a second ratio of time (minutes).

[pic] = [pic] [pic][pic] = [pic]

Using letters for the variables

x1 = y2 The subscripts remind us that x1 and y1 go together,

x2 y1 x2 and y2 do also. As in the example, x1 and y1

refer to the bike speed (or rate of speed) and the

bike time.

Compare this with two variables that are in direct proportion

x1 = y1

x2 = y 2 Because it follows that x1y2 = x2y1 we may be casual as we said before and write

x1 = x2 When we cross multiply we get x1y2 = x2y1 just as

y1= y2 we did when we didn’t mix our x’s and y’s.

PRACTICE

If y varies inversely with x, find the missing value.

1. y = 6 when x = 8, y = ? when x = 2.

2. y = 24 when x = 6, y = 12 when x = ?

3. y = 1 when x = 1, y = [pic] when x = ?

4. Sue needs 2 cups of sugar for a recipe for 8 dozen cookies. How much sugar will she need for 6 dozen?

5. When Doug averages 45 mph driving to work it takes him 30 minutes. How long will it take if he is able to average 50 mph covering the same route?

6. If a 30-foot telephone pole casts a 24-foot shadow, how tall is the tree that casts a 31-foot shadow?

ANSWERS

PRACTICE PAGE 3 PRACTICE PAGE 5

1. 35 6. 72 1. 24

2. 25 7. $3.60 2. 12

3. 2 8. 40 in. 3. 3

4. 5 9. 22 lb. 4. 1[pic] cups

5. 15 10. 26[pic] capsules 5. 27 minutes

6. 38 [pic] feet

OPTIONAL

An alternate approach to the understanding of proportion puts the emphasis on the fact that proportion expresses the way quantities vary.

Consider the problem: Suppose y varies directly as x. If y is 8 when x is 2 what is y when x is 3? To say that y varies directly as x is to say that y is a multiple of x. What multiple? We may express y as a multiple of x symbolically by showing that y is some number multiplied by x. We represent that number by k; then y = kx.

If y=8 when x=2, we may write 8 = k 2 and see that k=4.

What is y when x =3? It follows that y = 4 3 and thus y = 12.

In general, y1 = k x1 and y2 = kx2 may represent two pairs of numbers that are in the same ratio and therefore form a proportion where x1 = y1. With a little algebra, we can

x2 y2

show that this proportion follows logically if y1 = k x1 and y2 = k x2 .

Solve each of the last two equations for k:

y1 = k x1 y2 = k x2

y1 = k y2 = k

x1 x2

The result, y1 = k = y2 indicates that the direct proportion does follow logically.

x1 x2

Often, you can solve direct variation problems mentally, especially when k, the number indicating what to multiply x by to find y, is a whole number. For example,

If y is 3 when x is 1 then y is 3 times x; so if x is 6, to find y multiply the 6 by 3 and y is 18.

Try a few.

1. If y is 10 when x is 2 then y = _____ when x = 4.

2. If y is 5 when x is 15 then y = _____when x = 9.

3. If y is 12 when x is 4 then y = 15 when x = _____.

Now we’ll look at inverse variation. If direct variation is represented by y = kx, it would seem to make sense to represent inverse variation by y = k [pic] since [pic] is the inverse of x. y = k [pic] may be written y = [pic] .

Again using subscripts to indicate two pairs of numbers, now in inverse variation, we have y1 = [pic] and y2 = [pic] Solving each for k:

y1 x1 = k y2x2 = k ( y1x1 = y2x2 If we apply some more algebra

y1x1 = y2x2 x1 = y2 Here we are, back to our first form

x2y1 x2y1 x2 [pic] y1 of the inverse proportion.

EXERCISES

Solve each proportion for the unknown.

1. [pic] = [pic] 2. [pic] = [pic] 3. [pic] = [pic]

4. [pic] = [pic] 5. [pic] = [pic] 6. [pic] = [pic]

4.

7. A model clipper ship is constructed on the scale of 10 ft. to 1[pic]in. on the model. How high would the mast be on the model if it is actually 75 feet high?

8. It takes 5 painters 2 days to paint a house. How long would it take 10 painters to paint the same house?

9. Jones and Smith are partners who divide their profits in the ratio of 3:5. Find Jones’ profit if Smith receives $15,000.

10. Jean can assemble 12 car parts in 60 minutes. How many minutes would she need to assemble 15 car parts?

11. The time required to travel one lap on a racetrack is inversely proportional to a car’s average speed. A car averaging 90 mph takes 2 minutes to complete one lap. How long will it take to complete one lap at 120 mph?

12. If a record store advertises “4 records for $17.00,” how many records can you buy for $25.50?

ANSWERS

Page 6 Try a Few Page 7 Exercises

1. 20 1. 98 7. 11[pic] feet

2. 3 2. 36 8. 1 day

3. 5 3. 25 9. $9000

4. 80 10. 75 car parts

5. 19[pic] 11. 1[pic] minutes

6. 16 12. 6 records

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