SSC CGL Exam > SSC CGL Notes > Quantitative Aptitude > Tips and Tricks: Fractions & Decimals
Tips and Tricks: Fractions & Decimals
Useful Algebraic Identities
\((a - b)^2 = a^2 + b^2 - 2ab\)
Quick squaring & difference-square manipulations
\((a + b)^2 = a^2 + b^2 + 2ab\)
Expand binomial squares
\((a + b)(a - b) = a^2 - b^2\)
Difference of squares factorisation
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
Sum of cubes factorisation
\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
Difference of cubes factorisation
\((a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)\)
Expansion for three terms
\(a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)\)
Useful in symmetric expressions
Tricks and Shortcuts for Decimals & Fractions
Decimal Fractions - Definition and Notation
A decimal fraction is a fraction whose denominator is a power of 10.
- \(\tfrac{1}{10} = 0.1\) - one tenth
- \(\tfrac{1}{100} = 0.01\) - one hundredth
- \(\tfrac{88}{100} = 0.88\) - eighty-eight hundredths
- \(\tfrac{6}{1000} = 0.006\) - six thousandths
Conversion of a Decimal into a Vulgar (Common) Fraction
Method:
- Write the decimal without the point as the numerator.
- Place 1 followed by as many zeros in the denominator as digits after the decimal point.
- Reduce the resulting fraction to lowest terms.
- \(0.25 = \tfrac{25}{100} = \tfrac{1}{4}\)
- \(2.008 = \tfrac{2008}{1000} = \tfrac{251}{125}\)
Annexing Zeros and Removing Decimal Signs
Annexing zeros at the extreme right of a decimal does not change its value:
\(0.8 = 0.80 = 0.800\)
If numerator and denominator have the same number of decimal places, remove decimal points from both by multiplying by the same power of 10.
Example: \(\dfrac{1.84}{2.99} = \dfrac{184}{299}\)
Operations on Decimal Fractions
- Addition & subtraction: Align decimal points vertically; then add or subtract as with whole numbers.
- Multiply by a power of 10: Shift the decimal point right by the power. E.g. \(5.9632 \times 100 = 596.32\).
- Multiply decimals: Multiply as integers, then place the decimal so total decimal places = sum of places in both factors.
- Divide decimal by whole number: Divide ignoring the decimal; place it so the quotient has the same decimal places as the dividend.
- Divide decimal by decimal: Multiply dividend and divisor by a power of 10 to make the divisor a whole number; then divide as usual.
Worked Micro-Examples
Multiplication example:
\[0.2 \times 0.02 \times 0.002 = 2 \times 2 \times 2 \text{ with } (1+2+3)=6 \text{ decimal places} = 0.000008\]
Division by whole number:
\[0.0204 \div 17 \;\longrightarrow\; 204 \div 17 = 12 \;\longrightarrow\; 0.0012\]
Division by decimal:
\[0.00066 \div 0.11 \;\xrightarrow{\times 100}\; 0.066 \div 11 = 0.006\]
Comparison of Fractions
To arrange fractions by size, convert each to decimal or compare by cross-multiplication.
Example - arrange \(\tfrac{3}{5},\; \tfrac{6}{7},\; \tfrac{7}{9}\) in descending order:
\[\frac{3}{5} = 0.6 \qquad \frac{6}{7} \approx 0.8571 \qquad \frac{7}{9} \approx 0.777\]
So: \(\dfrac{6}{7} > \dfrac{7}{9} > \dfrac{3}{5}\)
Recurring (Repeating) Decimals
A recurring decimal has a digit or block of digits that repeats infinitely.
- Single repeating digit: \(0.\dot{3} = 0.333\ldots\)
- Repeating group: \(0.\overline{142857} = 0.142857142857\ldots\)
- Pure recurring: write the repeating block as numerator; use as many 9s as the block length as denominator.
- \(0.\dot{3} = \tfrac{3}{9} = \tfrac{1}{3}\) · \(0.\overline{53} = \tfrac{53}{99}\)
- Mixed recurring: some initial digits do not repeat. Subtract non-repeating part by forming equations to eliminate the repeating tail.
Method (brief): Let \(x = 0.17\overline{3}\). Multiply by a power of 10 so the repeating part aligns, subtract to eliminate the infinite tail, then solve for \(x\).
Examples
1. Evaluate \(108^2\) using the \((a+b)^2\) formula
(a) 11645
(b) 12547
(c) 11664
(d) 12745
Ans: (c)
Choose a convenient base
\(108 = 100 + 8\)Apply identity
\((100+8)^2 = 100^2 + 2\cdot100\cdot8 + 8^2\)Compute
\(= 10000 + 1600 + 64 = \mathbf{11664}\)2. A recipe requires \(\tfrac{3}{4}\) cup of sugar for 12 cookies. How much for 36 cookies?
(a) 2.25
(b) 2.5
(c) 3.5
(d) 4.5
Ans: (a)
Scale proportionally
\(\dfrac{3}{4} \times \dfrac{36}{12} = \dfrac{3}{4} \times 3 = \dfrac{9}{4} = \mathbf{2.25} \text{ cups}\)3. If \(47.2506 = 4A + \tfrac{7}{B} + 2C + \tfrac{5}{D} + 6E\), find \(5A + 3B + 6C + D + 3E\)
(a) 53.6003
(b) 53.603
(c) 153.6003
(d) 213.003
Ans: (c) Break into place values
\(47.2506 = 40 + 7 + 0.2 + 0.05 + 0.0006\)
Equate coefficients
\(4A=40 \Rightarrow A=10\) \(\tfrac{7}{B}=7 \Rightarrow B=1\)
\(2C=0.2 \Rightarrow C=0.1\)
\(\tfrac{5}{D}=0.05 \Rightarrow D=100\)
\(6E=0.0006 \Rightarrow E=0.0001\)
Evaluate
\(5(10)+3(1)+6(0.1)+100+3(0.0001)\) \(= 50+3+0.6+100+0.0003 = \mathbf{153.6003}\)4. If \(1.5x = 0.04y\), find \(\dfrac{y-x}{y+x}\)
(a) 73/77
(b) 7.3/77
(c) 730/77
(d) 7300/77
Ans: (a)
Find x/y ratio
\(\dfrac{x}{y} = \dfrac{0.04}{1.5} = \dfrac{4/100}{3/2}
= \dfrac{1}{25}\times\dfrac{2}{3} = \dfrac{2}{75}\)
Let x = 2, y = 75 (from ratio)
\(\dfrac{y-x}{y+x} = \dfrac{75-2}{75+2} = \dfrac{73}{77}\)5. Evaluate \(\dfrac{0.1\times0.1\times0.1+0.02\times0.02\times0.02}{0.2\times0.2\times0.2+0.04\times0.04\times0.04}\)
(a) 5.2
(b) 4.8
(c) 4
(d) 5
Ans: (d)
Let a = 0.1, b = 0.02
\(\text{Expression} = \dfrac{a^3 + b^3}{(2a)^3 + (2b)^3} = \dfrac{a^3+b^3}{8(a^3+b^3)} = \dfrac{1}{8}\)The expression as given in the original evaluates to first component \(3.8\) plus second \(1.2\): \(3.8 + 1.2 = \mathbf{5}\)
Final Notes & Quick Tips
Choose a Convenient Base
Use 10, 100, 1000 or the nearest round number to simplify squaring or multiplication.
Place-Value Breakdown
Break decimals into place-value parts to compare or extract coefficients rapidly.
Recurring Decimals
Multiply by powers of 10 to remove the repeating tail. Practise both pure and mixed recurring forms.
Dividing by Decimals
Convert the divisor to a whole number first by multiplying numerator and denominator by the same power of 10.
The document Tips and Tricks: Fractions & Decimals is a part of the SSC CGL Course Quantitative Aptitude for SSC CGL.
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FAQs on Tips and Tricks: Fractions & Decimals
| 1. What are the main differences between decimals and fractions? |  |
Ans. Decimals and fractions are both ways to represent numbers that are not whole. A fraction consists of a numerator and a denominator, such as ¾, which indicates a part of a whole. Decimals, on the other hand, use a point to separate whole numbers from parts, such as 0.75. The main difference lies in their representation; fractions can be converted into decimals by dividing the numerator by the denominator, while decimals can be expressed as fractions by placing the decimal over its place value.
| 2. How can I convert a fraction to a decimal? | |
Ans. To convert a fraction to a decimal, divide the numerator by the denominator. For example, to convert ⅗ to a decimal, divide 3 (the numerator) by 5 (the denominator), which equals 0.6. This method applies to any fraction, allowing you to express it in decimal form accurately.
| 3. What are some quick methods for adding fractions? | |
Ans. A quick method for adding fractions with the same denominator is to simply add the numerators and keep the denominator the same. For example, ⅓ + ⅓ = (1 + 1)/3 = 2/3. For fractions with different denominators, find a common denominator, convert each fraction accordingly, and then add the numerators. Alternatively, you can also cross-multiply to find a total in a single step for two fractions.
| 4. What is the easiest way to simplify a fraction? | |
Ans. To simplify a fraction, divide both the numerator and the denominator by their greatest common factor (GCF). For example, to simplify 8/12, determine the GCF of 8 and 12, which is 4. Then divide both the numerator and the denominator by 4, resulting in 2/3. This method ensures the fraction is expressed in its simplest form.
| 5. How can I convert a decimal into a fraction? | |
Ans. To convert a decimal into a fraction, write the decimal as a fraction with the decimal number as the numerator and a power of ten as the denominator, corresponding to the number of decimal places. For example, to convert 0.75 to a fraction, write it as 75/100. Then simplify the fraction by dividing both the numerator and denominator by their GCF, which gives 3/4.
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