Numbers Decimal And Fraction Formulas

Number, Decimal and Fraction Formulas 

1. (a + b)(a – b) = (a² – b²)

2. (a + b)² = (a² + b² + 2ab)

3. (a – b)² = (a² + b² – 2ab)

4. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

5. (a³ + b³) = (a + b)(a² – ab + b²)

6.(a³ – b³) = (a – b)(a² + ab + b²)

7. (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)

8. when a + b + c = 0, then a³ + b³ + c³ = 3abc

Conversion of a Decimal into Vulgar Fraction:

Put 1 in the denominator under the decimal point and annex with it as many zeros as in the number of digits after the decimal point.

• Now, remove the decimal point and reduce the fraction to its lowest terms.

For Ex- 0.45 =  \frac{45}{100} \ =  \frac{9}{25} \

Annexing Zeros and Removing Decimal Signs:

Applying zeros to the extreme right of a decimal fraction does not change its value. Thus, 0.5 = 0.50 = 0.500, etc.

• If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.

For Ex-  \frac{5.879}{4.856} \ =   \frac{5879}{4856} \ .

Operations Formulas For Number, Decimals and Fractions

1. Addition and Subtraction of Decimal Fractions: The given numbers are placed under each other that the decimal points lies in one column. The numbers are so arranged that can now be added or subtracted in the usual way.
2. Multiplication of a Decimal Fraction By a Power of 10: Shift the decimal point to the right by as many places as is the power of 10.

   Thus, 5.9632 x 100 = 596.32;   0.073 x 10000 = 730.

3. Multiplication of Decimal Fractions: Multiply the given numbers considering them without decimal point. Now, in the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers.

   Suppose we have to find the product (.2 x 0.02 x .002).

   Now, 2 x 2 x 2 = 8. Sum of decimal places = (1 + 2 + 3) = 6.

    .2 x .02 x .002 = .000008

4. Dividing a Decimal Fraction By a Counting Number: Divide the given number without considering the decimal point, by the given counting number. Now, in the quotient, put the decimal point to give as many places of decimal as there are in the dividend.

   Suppose we have to find the quotient (0.0204 ÷ 17). Now, 204 ÷ 17 = 12.

   Dividend contains 4 places of decimal. So, 0.0204 ÷ 17 = 0.0012

5. Dividing a Decimal Fraction By a Decimal Fraction: Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number.

For Ex- \frac{0.0066}{0.12} \ x  \frac{0.0066 * 100}{0.12 * 100} \ =  \frac{0.0066}{0.12} \ = \frac{0.66}{12}\

Number, Decimal and Fraction Formulas 

Suppose some fractions are to be arranged in ascending or descending order of magnitude, then convert each one of the given fractions in the decimal form, and arrange them accordingly.

For Ex- Arrange the fractions, in descending order.
 \frac{3}{7} \ ,  \frac{5}{21} \ ,  \frac{9}{7} \

Now,  \frac{3}{7} \ = 0.42,  \frac{5}{21} \ = 0.23,  \frac{9}{7} \ = 1.28

Since, 1.28 is greater. So, 1.28>0.42>0.23.
Therefore,  \frac{3}{7} \ >  \frac{5}{21} \ >  \frac{9}{7} \ .

Recurring Decimal:

If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal.

In a recurring decimal, if a single figure is repeated, then it is expressed by putting a dot on it. If a set of figures is repeated, it is expressed by putting a bar on the set.

\frac{1}{3}=0.\overline{33} , \frac{22}{7} \ = 3.142857142857…….. =  \frac{}{}3.\overline{142857}

Pure Recurring Decimal

A decimal fraction, in which all the figures after the decimal point are repeated, is called a pure recurring decimal.

Converting a Pure Recurring Decimal into Vulgar Fraction

Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures.

For Ex- 0.5 = \frac{5}{9}= 0.\overline{53} ;  \frac{53}{99} \ .

Mixed Recurring Decimal

A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal.
For Ex- 0.175555 = \frac{}{} 0.17\overline{3}