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
Place the digit 1 in the denominator beneath the decimal point and add a corresponding number of zeros to it, equal to the count of digits following the decimal point.
Now, remove the decimal point and reduce the fraction to its lowest terms.
For Ex 0.45 = 45/100 = 9/25
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 5.879/4.856 = 5879/4856.
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.
A decimal fraction is termed a recurring decimal when a digit or a group of digits is consistently repeated. In the case of a recurring decimal, a single digit repetition is denoted by placing a dot on it, while a repeated set of digits is represented by placing a bar over the set.
A pure recurring decimal is a decimal fraction where every digit following the decimal point is repeated.
Express the recurring digits by writing them once in the numerator, and use a corresponding number of nines in the denominator equal to the count of repeating figures.
A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal.
Example 1: √196 × √144 × 20% of 700 = ? + 1265
(a) 22255
(b) 22266
(c) 22588
(d) 25874
Ans: (a)
√196 × √144 × 20% of 700 = ? + 1265
14 × 12 × 140 = ? + 1265
168 × 140 = ? + 1265
23520 = ? + 1265
? = 23520 – 1265
? = 22255
Example 2: 25% of 30% of 850 + 5 × 76 = ?
(a) 441.75
(b) 443.75
(c) 441.57
(d) 443.57
Ans: (b)
25% of 30% of 850 + 5 × 76
= 25/100 * 30/100 * 850 + 5 * 76
= 443.75
Example 3: (32/100) × 750 – ? = (14/100) × 540
(a) 152.6
(b) 132.6
(c) 146.6
(d) 164.4
Ans: (d)
(32/100) × 750 – ? = (14/100) × 540
240 – x = 75.6
x = 240 – 75.6
x = 164.4
Example 4: 4100 + 13.952 – ? = 3764.002
(a) 747.095
(b) 247.752
(c) 347.932
(d) 349.95
Ans: (d)
Let 4100 + 13.952 – x = 3764.002.
Then x = (4100 + 13.952) – 3764.002
= 4113.952 – 3764.002 = 349.95.
Example 5: What is the sum of the decimal fractions 25/100 and 30/100?
(a) 55/100
(b) 65/100
(c) 75/100
(d) 85/100
Ans: (a)
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
(25/100) + (30/100) = 55/100.
Hence, the sum of the decimal fractions 25/100 and 30/100 is 55/100.
314 videos170 docs185 tests

1. What are some important formulas for numbers, decimals, and fractions? 
2. How do you convert decimals to fractions? 
3. What is the difference between a fraction and a decimal? 
4. How do you add or subtract fractions? 
5. How do you multiply or divide fractions? 
314 videos170 docs185 tests


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