Number, Decimal, and Fraction Formulas
Basic Formulas
- (a + b)(a – b) = (a² – b²)
- (a + b)² = (a² + b² + 2ab)
- (a – b)² = (a² + b² – 2ab)
- (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
- (a³ + b³) = (a + b)(a² – ab + b²)
- (a³ – b³) = (a – b)(a² + ab + b²)
- (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
- 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.
Operations Formulas
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.
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.
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.
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.
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.
Arranging Fractions in Ascending or Descending Order
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.
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.
Question and Answers
Question 1:
4100 + 13.952 – ? = 3764.002
Options:
A. 747.095
B. 247.752
C. 347.932
D. 349.95
Answer: Option D
Explanation: Let 4100 + 13.952 – x = 3764.002. Then x = (4100 + 13.952) – 3764.002 = 4113.952 – 3764.002 = 349.95.
Question 2:
What is the sum of the decimal fractions 25/100 and 30/100?
Options:
A. 55/100
B. 65/100
C. 75/100
D. 85/100
Solution: Given decimal fractions: 25/100 and 30/100. As the denominators are the same in both decimal fractions, we can directly add the numerators. Thus, (25/100) + (30/100) = (25 + 30)/100 = 55/100. Hence, the sum of the decimal fractions 25/100 and 30/100 is 55/100.