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# COMPARING FRACTIONS

Lesson 23  Section 2

Back to Section 1

The ratio of two fractions

We saw in Lesson 20 that when two fractions have equal denominators, then the larger the numerator, the larger the fraction.

 25 is larger than 15 .
 But what specifically is the ratio of 25 to 15 ?
 25 is to 15 as  2 is to 1.
 25 is two times 15 .

In other words:

Fractions with equal denominators are in the same ratio
as their numerators.

 25 is to 35 as  2 is to 3.
 25 is two thirds of 35 .

 4. How can we know the ratio of any two fractions? By "cross-multiplying."
 Example 1. 23 is to 58 as 2 × 8 is to 3 × 5 as 16 is to 15.

Why?  Because 16 and 15 are the numerators we would get if we

 expressed 23 and 58 with the common denominator 24:

as

 1624 is to 1524 .

The numerators 16 and 15 were obtained by "cross-multiplying."

 Upon cross-multiplying, we also see that 23 is larger than 58 .

For, 16 is larger than 15.

 Example 2.   Which is larger, 47 or 59 ?

 47 is to 59

as

36  is to  35.

Now, 36 is larger than 35.  Therefore,

 47 is larger than 59 .

Note:  We must begin multiplying with the numerator on the left:

4 × 9.

 Example 3. 14 is to 12 as which whole numbers?

 14 is to 12

as

2  is to  4.

That is,

 14 is half of 12 .

Example 4.   What ratio has 2½ to 3?

 Answer.   First, express 2½ as the improper fraction 52 .  Then, treat the

whole number 3 as a numerator, and cross-multiply:

 52 is to 3  as  5 is to 6.

2½ is five sixths of 6.

 Equivalently, since  3 = 62 (Lesson 21, Question 2), then
 52 is to 62 as  5 is to 6.

In general:

To express the ratio of a fraction to a whole number,
multiply the whole number by the denominator.

 67 is to 3  as  6 is to 21.

For an application of this, see Lesson 26.

 Example 5.   On a map, 34 of an inch represents 60 miles.  How many

miles does 2 inches represent?

Solution.  Proportionally,

 34 of an inch is to 2 inches  as  60 miles is to ? miles.
 What ratio has 34 to  2?
 34 is to 2  as  3 is to 8.

Therefore:

3 is to 8 as  60 miles is to ? miles.

Since  20 × 3 = 60,  then 20 × 8 = 160 miles.

The theorem of the same multiple.

Or, inversely,

 2 inches is to 34 of an inch  as  8 is to 3.

8 is two and two thirds times 3.

(Lesson 18, Example 5.)  Therefore:

 Two and two thirds times 60 = Two times 60 + two thirds of 60 Lesson 16 = 120 + 40 = 160 miles.

More than or less than ½

 5. How can we know whether a fraction is more than or less than ½? If the numerator is more than half of the denominator, then the fraction is more than ½. While if the numerator is less than half of the denominator, the fraction is less than ½.

 48 is equal  to 12 , because 4 is half of 8.  Therefore, 58 is more than 12 ,
 because 5 is more than half of 8;  while 38 is less than 12 , because 3 is less

than half of 8.

 Example 6.   Which is larger, 7 12 or 9 20 ?
 Answer. 7 12 .  Because 7 is more than half of 12, while 9 is less than half

of 20.

 Example 7.   Which is larger, 1121 or 1225 ?
 Answer. 1121 .  Because 11 is more than half of 21 (which is 10½); while 12 is

less than half of 25 (which is 12½).  (Lesson 15, Question 7.)

We could make these comparisons for any ratio of the terms.  For example, we could know that

 5 15 is larger than 6 21 .

Because 5 is a third of 15, but 6 is less than a third of 21 (which is 7).

Example 8   Which is the largest number?

 3 10 58 12 27 59

Answer.  First, let us examine the list to see if there are numbers less than ½ or greater than ½.  We may eliminate any numbers less than (or equal to) ½.

 Thus we may eliminate 3 10 , 12 , and 27 .
 We are left with 58 and 59 .

Since the numerators are the same (Lesson 20, Question 11), we

 conclude that the largest number is 58 .

Example 9.   Which is the largest number?

 59 25 6 11
 Answer.  We may eliminate 25 because it is less than ½, while the others

are greater. Which is larger, then,

 59 or 6 11 ?

On cross-multiplying, we have 5 × 11 versus 9 × 6.  And

55 is greater than 54.

Therefore,

 59 is greater than 6 11 .

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