4
Fractions
WE USE THE NATURAL NUMBERS for counting, but we cannot use them for measuring. For that, we have to invent the fractions.
Let us begin by observing that we know a number according to how it
is related to 1. What is our understanding of "2" ? It is twice as much as 1. What is "3"? It is three times more than 1.
Every number has a ratio to 1. It is according to that ratio that we know each number.
Since we measure things that are continuous, let us now think of 1
as a continuous unit, and let us imagine that we have to invent a numeral -- a symbol, a "number" -- for that point on the line which is half of 1.
What symbol should we invent?
| The symbol invented, of course, is 1 over 2: | 1 2 |
. But why? |
Because of the ratio of 1 to 2. Since 1 is one half of 2, then the number
| we write in this way, | 1 2 |
, is one half of 1. We call it the number "one-half." |
In other words, the fractions are named (in English) according to the ratio of the numerator to the denominator.
| This number | 2 3 |
is called "two-thirds" because of the ratio of 2 to 3. |
2 is two thirds of 3. The fraction, moreover, has that same ratio to 1:
| 2 3 |
is to 1 as 2 is to 3. |
The fraction called "two-thirds" is two thirds of 1.
(Notice that we write the name of the fraction hyphenated, but not the name of the ratio. In this way we maintain the distinction between fractions and ratios. Fractions indicate ratios, namely, the ratio of the numerator to the denominator.)
In general, every fraction has the same ratio to 1 as the numerator has to the denominator.
| a b |
: 1 = a : b |
Problem 1. a) What number is the fourth part of 1? Write its symbol and also write its name in words.
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| 1 4 |
. One-fourth. |
b) When we speak of a fourth of 1, is 1 a continuous unit or discrete?
Continuous.
| c) Place | 1 4 |
on this number line. |
| Problem 2. Why is this number | 2 5 |
called "two-fifths"? |
Because of the ratio of 2 to 5. 2 is two fifths of 5.
Problem 3. What ratio has each number to 1?
| a) | 1 2 |
is one half of 1. | b) | 1 3 |
is one third of 1. | |
| c) | 2 3 |
is two thirds of 1. | d) | 3 4 |
is three fourths of 1. | |
| e) | 4 5 |
is four fifths of 1. | f) | 5 8 |
is five eighths of 1. | |
Problem 3.
a) What is a proper fraction?
A fraction that is less than 1.
b) How can we recognize a proper fraction?
The numerator is less than the denominator.
c) What is a mixed number?
A whole number plus a proper fraction. For example, 4½.
Problem 4. Continuous versus discrete Answer with a mixed number, or with a whole number and a remainder, whichever makes sense.
a) It takes three yards of material to make a skirt. How many skirts can
a) you make from 25 yards?
8 skirts. 1 yard will remain. 8 1/3 skirts makes no sense. Skirts are discrete.
b) You are going on a journey of 25 miles, and you have gone a third of
b) the distance. How far have you gone?
8 1/3 miles. Here, the mixed number makes sense. We need mixed numbers for measuring.
Problem 5. What is an improper fraction.
A fraction greater than or equal to 1.
Recall from arithmetic that we can always express a mixed number as an improper fraction.
Problem 6. Express 6½ as an improper fraction.
| 6½ | = | 13 2 |
Problem 7. Complete each proportion with natural numbers.
| a) | 1 2 |
: 1 = 1 : 2 | b) | 
| 2 : 3 | ||
| c) | 3 2 |
: 1 = 3 : 2 | d) | 5¼ : 1 = | 21 4 |
: 1 = 21 : 4 | |
e) .49 : 1 = 49 : 100. Multiply both terms by 100.
f) 2.5 : 1 = 25 : 10. Multiply both terms by 10.
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