Common Fractions and Mixed Numbers
We use fractions when we need to consider a part of a whole object. A common fraction consists of three components:
The denominator shows in how many equal parts a whole object is divided, the numerator shows how many of those parts are taken in the consideration.
If the numerator of a fraction is less than its denominator, the fraction is called a proper fraction: , ... The value of a proper fraction is always less than 1. If the numerator of a fraction is greater than its denominator, the fraction is called an improper fraction:, ... The value of an improper fraction is always greater than 1. An improper fraction can be always expressed as a mixed number containing a whole part and a fractional part:, ...
To convert a mixed number to an improper fraction, multiply the whole part by the denominator of the fractional part, add the numerator and place the result in the numerator of the resulting fraction; the denominator will stay the same.
To convert an improper fraction into a mixed number, divide its numerator by the denominator. The quotient will give the whole part of the mixed number and the remainder - the numerator of the fractional part; the denominator will stay the same.
The main property of fraction: both the numerator and the denominator can be multiplied or divided by the same number without changing the fraction value.
We can use this property for reducing a fraction by division of its numerator and denominator by their greatest common factor. The greatest common factor (GCF) of two numbers is the greatest number by which these two numbers are divisible.
A. . The GCF of 8 and 12 is 4. Divide the numerator and the denominator by 4:.
B. . The GCF of 7 and 35 is 7. Divide the numerator and the denominator by 7:.
C. . The GCF of 105 and 168 is 21. Divide the numerator and the denominator by 21:.
Multiplication and Division of Fractions
To multiply two fractions multiply their numerators and place the product in the numerator of the resulting fraction; multiply their denominators and place the product in the denominator of the resulting fraction. Before doing so, cross reduce (if possible) their denominators and their numerators. To multiply a whole number by a fraction rewrite the whole number with a denominator of 1.
A reciprocal of a fraction is a fraction with interchanged numerator and denominator:
and , and 6, ─13 and .
To divide one fraction by another replace a division sign with a multiplication sign and the second fraction with its reciprocal; then, perform multiplication.
In order to multiply or divide mixed numbers they must be converted into improper fractions.
Addition and Subtraction of Fractions
To add two fractions with alike denominators add their numerators. To add two fractions with unlike denominators write the least common denominator (the least common multiple) under the long fraction line. Divide this common denominator by the denominator of each fraction and multiply by the numerator of this fraction. Place the results above the fraction line and then add them. The least common multiple of two numbers is the smallest number divisible by both of these numbers.
In the same way one fraction can be subtracted from another fraction.
To add mixed numbers add separately their whole parts and separately their fractional parts, then combine these parts.
In the same simple way one mixed number can be subtracted from another mixed number if this subtraction does not create negative whole parts or negative fractional parts.
To subtract a fraction or a mixed number from a whole number borrow 1 from the whole number and express this 1 with denominator the same as denominator of the fractional part of the mixed number. Remember 1 can be expressed as any number divided by itself.
This technique can be used if subtraction of fractional parts of mixed numbers results in a negative value.
To convert a decimal to a fraction or a mixed number write its fractional part as a fraction with a power of 10 denominator (10, 100, 1000 and so on). If necessary, reduce the fraction.
To convert a fraction or a mixed number to a decimal divide its numerator by the denominator. If a denominator contains only factors of 2 and 5 expect a terminating decimal, if any other factors – expect a decimal with repeating digits.
To multiply a decimal by a power of 10 move the decimal point to the right, to divide by the power of 10 – to the left. The number of decimal places to move equals to the number of zeros after 1 or to the exponent of 10.
A percent is 1/100 part of a whole unit. To convert a decimal to a percent equivalent multiply the decimal by 100. To convert percents into a decimal divide percents by 100.
A. 56% = 0.56.
B. 0.248 = 24.8%.
C. 12.5 = 1250%
D. 0.089 = 0.00089
The basic formula for percentage problems:
where A is the amount (defining word – is), P is the percentage (must be converted to a decimal), T is the total (defining word – of). This formula has three variables; if two of them are given, the third can be found.
A. 9 is 15% of what? A=9, P=0.15, T=? 9 = 0.15 x T; T = 60.
B. What percent is 3.25 of 26? T=26, A=3.25, P=? 3.25 = P x 26; P=0.125=12.5%.
C. The price of a computer is $600. If the sales tax rate is 9.65%, find the amount of sales tax.
T=600, P=0.0965, A=? A = 0.0965 x 600; A=$57.90.
D. What is 65% of 120? T=120, P=0.65, A=? A = 0.65x120; A = 78. If his commission rate is 6%, find the price of the property.
A=120,000, P=0.06, T=? 120000 = 0.06 x T; T=$2,000,000.
E. A family is going for the 1069-mile trip. If in the first day they covered 626 miles, what percent of the trip they made?
T=1069, A=626, P=? 626 = P x 1069;
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