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
| 2 3 | is to | 5 8 | as | 16 is to 15, |
| then since 16 is larger than 15, you would know that | 2 3 | is larger than | 5 8 | . |
Now we saw in Lesson 20 that when two fractions have equal denominators, then the larger the numerator, the larger the fraction.
| 2 5 | is larger than | 1 5 | . |
| But what specifically is the ratio of | 2 5 | to | 1 5 | ? |
| 2 5 | is to | 1 5 | as 2 is to 1 . |
| 2 5 | is two times | 1 5 | . |
Fractions with equal denominators are in the same ratio
as their numerators
| 2 5 | is to | 3 5 | as 2 is to 3 . |
When fractions do not have equal denominators, then we can know their ratio -- we can compare them -- by cross-multiplying. Because that gives the numerators if we had expressed them with equal denominators.
| 4. | How can we compare fractions by cross-multiplying? |
| Cross-multiply and compare the numerators. |
| Example 1 . | 2 3 | is to | 5 8 |
| as | 2 × 8 | is to | 3 × 5 |
| as | 16 | is to | 15. |
| 16 and 15 are the numerators we would get if we expressed | 2 3 | and | 5 8 |
| 16 24 | is to | 15 24 | . |
| And since 16 is larger than 15, we would know that | 2 3 | is larger than | 5 8 | . |
| Example 2 . Which is larger, | 4 7 | or | 5 9 | ? |
Answer . On cross-multiplying,
| 4 7 | is to | 5 9 |
36 is larger than 35. Therefore,
| 4 7 | is larger than | 5 9 | . |
Note: We must begin multiplying with the numerator on the left:
| Example 3 . | 1 4 | is to | 1 2 | as which whole numbers? |
Answer . On cross-multiplying,
| 1 4 | is to | 1 2 |
| 1 4 | is half of | 1 2 | . |
Example 4. What ratio has 2½ to 3?
| Answer. First, express 2½ as the improper fraction | 5 2 | . Then, treat the |
whole number 3 as a numerator, and cross-multiply:
| 5 2 | is to 3 as 5 is to 6 . |
| Equivalently, since 3 = | 6 2 | (Lesson 21, Question 2), then |
| 5 2 | is to | 6 2 | as 5 is to 6 . |
To express the ratio of a fraction to a whole number,
multiply the whole number by the denominator.
| 6 7 | is to 3 as 6 is to 21. |
For an application of this, see Lesson 26.
| Example 5. On a map, | 3 4 | of an inch represents 60 miles. How many |
miles does 2 inches represent?
| 3 4 | of an inch is to 2 inches as 60 miles is to ? miles. |
| What ratio has | 3 4 | to 2? |
| 3 4 | is to 2 as 3 is to 8. |
3 is to 8 as 60 miles is to ? miles.
Since 20 × 3 = 60 , then 20 × 8 = 160 miles.
8 is to 3 as ? miles is to 60 miles.
8 is two and two thirds times 3.
(Lesson 18, Example 5.) Therefore, the missing term will be
| 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. |
| 4 8 | is equal to | 1 2 | , because 4 is half of 8. Therefore, | 5 8 | is more than | 1 2 | , |
| because 5 is more than half of 8; while | 3 8 | is less than | 1 2 | , because 3 is less |
| 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 |
| Example 7. Which is larger, | 11 21 | or | 12 25 | ? |
| Answer . | 11 21 | . Because 11 is more than half of 21 (which is 10½); while 12 |
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
| 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 | 5 8 | 1 2 | 2 7 | 5 9 |
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.
| And so we may eliminate | 3 10 | , | 1 2 | , and | 2 7 | . |
| We are left with | 5 8 | and | 5 9 | . |
Since the numerators are the same (Lesson 20, Question 11), we
| conclude that the largest number is | 5 8 | . |
Example 9. Which is the largest number?
| 5 9 | 2 5 | 6 11 |
| Answer . We may eliminate | 2 5 | because it is less than ½, while the others |
are greater. Which is larger, then,
| 5 9 | or | 6 11 | ? |
On cross-multiplying, we have 5 × 11 versus 9 × 6. And
55 is greater than 54.
| 5 9 | is greater than | 6 11 | . |
Please "turn" the page and do some Problems.
Continue on to the next Lesson.
Copyright © 2021 Lawrence Spector