Fractions and Decimals
• When a whole is divided into
n equal parts, each part is called one nth of
the whole, written
1/n.
For example:
1. If a pizza is cut (divided) into 8 equal slices, each slice is one eighth
(1/8) of the pizza;
2. A day is divided into 24 equal hours, so an hour is one twenty-fourth
(1/24) of a day and
3. An inch is one twelfth
( 1/12) of a foot.
4. If one works for 8 hours a day, he works eight twenty-fourth
( 8/24) of a day.
5. If a hockey stick is 40 inches long, it measures forty twelfths
(40/12) of a foot.
• The numbers such as
1/8, 1/24, 8/24 and 40/12 in which one integer is written over the second integer, are called
fractions. The center line is called the
fraction bar. The number above the bar is called the
numerator, and the number below the bar is called
denominator.
• The denominator of a fraction can never be
0.
• A fraction, such as 1/24 , in which the denominator is greater than numerator, is known as a
proper fraction. Its value is less than one.
• A fraction, such as 40/12, in which the denominator is less than numerator, is known as an
improper fraction. Its value is greater than one.
• A fraction, such as, 12/12 in which the denominator is equal to the numerator, is also known as an
improper fraction. But, Its value is one.
• Every fraction can be expressed in decimal form (or as a whole number) by dividing the number by the denominator.
3/10= 0.3, 3/4=0.75, 8/8=1, 48/16=3, 100/8=12.5
• Unlike the examples above, when most fractions are converted to decimals, the division does not terminate, after 2 or 3 or 4 decimal places; rather it goes on forever with some set of digits repeating it.
2/3= 0.66666..., 3/110.272727..., 5/12=0.416666..., 117/15.133333...
• To compare two decimals,
follow these rules:
o Whichever number has the greater number to the left of the decimal point is greater: since
11 > 9, 11.0001 > 9.8965 and since
1 > 0, 1.234 > .8. (Recall that if a decimal is written without a number on left of decimal point, you may assume that a 0 is there, so,
.8 = 0.8).
o If the numbers to the left of the decimal point are equal, proceed as follows:
• If the numbers do not have the same number of digits to the right of the decimal point, add zeroes to the end of the shorter one to make them equal in length.
• Now compare the numbers ignoring the decimal point.
• For example, to compare
1.83 and 1.823, add a
0 to the end of
1.83 forming 1.830. Now compare them, thinking of them as whole numbers without decimal point: since
1830 > 1823, then 1.830 >1.823.
•
There are two ways to compare fractions:
o
Convert them to decimals by dividing, and use the method already described to compare these decimals. For example to compare
2/5 and 1/4 , convert them to decimals.
2/5 = 0.4 and 1/4 = 0.25 . Now, as 0
.4 > 0.25, 2/5 > 1/4.
o
Cross multiply the fractions. For example to compare
2/5 and 1/4, cross multiply:
2/5 × 1/4 Since
2×4 > 1×5 , then
2/5 > 1/4.
• While comparing the fractions, if they have
same denominators, the fraction with the larger numerator is greater. For example
3/5 > 2/5 .
• If the fractions have the
same numerator, the fraction with the
smaller denominator is greater. For example 3/5 > 3/10.
• Two fractions are called
equivalent fractions if both of them have same decimal value. For example,
1/2 = 5/10 as both of these are equal to
0.5.
• Another way
to check the equivalence of two fractions is to cross-multiply. If both of the products are same, the fractions are equivalent.
For Example,
to compare
2/5 with 6/15, cross-multiply. Since
2 × 15 = 6 × 5 , both of the fractions are equivalent.
• Every fraction can be reduced to lowest terms by dividing the numerator
and denominator by their
greatest common divisor (GCD). If the GCD is 1, the fraction is already in lowest terms. For example to reduce
10/15, divide both numerator and denominator by 5 (which is GCD of 10 and 15). This will reduce the fraction to
2/3.
•
To multiply two fractions, multiply their numerators and multiply their denominators. For example
3/5 × 4/7 = 3 × 4/ 5 × 7 = 12/35.
•
To multiply a number to a fraction, write that number as a fraction whose denominator is 1. For example
3/5 × 7: 3/5 × 7/1 = 3 × 7/5 × 1= 21/5.
• When a problem requires you to find the fraction of a number, multiply that fraction with the number. For example, to find two fifth
(2/5) of 200,
multiply: 2/5 × 200 = 2/5 × 200/1 = 400/5 = 80.
• The
reciprocal of a fraction a/b is another fraction b/a since
a/b × b/a=1.
•
To divide one fraction by the other fraction, multiply the reciprocal of divisor with the dividend. For example,
22/7 ÷ 11/7 = 22/ 7 × /11=2/1 =2.
• To
add or subtract the fractions with same denominator, add or subtract numerators and keep the denominator. For example
4/9 + 1/9 =5/9 and 4/9-1/9 = 3/9.
Continued...