COMPARING FRACTIONSLesson 23 Section 2 The ratio of two fractions To know the ratio of fractions, is to compare them. We are about to see: Fractions have the same ratio to one another as natural numbers. If you knew that
Now we saw in Lesson 20 that when two fractions have equal denominators, then the larger the numerator, the larger the fraction.
In other words: Fractions with equal denominators are in the same ratio
When fractions do not have equal denominators, then we can know their ratio  we can compare them  by crossmultiplying. Because that gives the numerators if we had expressed them with equal denominators. 



with the common denominator 24. as
Answer. On crossmultiplying,
as 36 is to 35. Now, 36 is larger than 35. Therefore,
Note: We must begin multiplying with the numerator on the left: 4 × 9.
Answer. On crossmultiplying,
as 2 is to 4. That is,
Example 4. What ratio has 2½ to 3?
whole number 3 as a numerator, and crossmultiply:
2½ is five sixths of 6.
In general: To express the ratio of a fraction to a whole number,
For an application of this, see Lesson 26.
miles does 2 inches represent? Solution. Proportionally,
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: 8 is to 3 as ? miles is to 60 miles. Now, 8 is two and two thirds times 3. (Lesson 18, Example 5.) Therefore, the missing term will be
More than or less than ½ 



than half of 8.
of 20.
is less than half of 25 (which is 12½). (Lesson 16, Question 8.) We could make these comparisons for any ratio of the terms. For example, we could know that
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?
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.
Since the numerators are the same (Lesson 20, Question 11), we
Example 9. Which is the largest number?
are greater. Which is larger, then,
On crossmultiplying, we have 5 × 11 versus 9 × 6. And 55 is greater than 54. Therefore,
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