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This is part of our
GRE Math Essentials project is the best complement to our
Official GRE Quant Book. It provides a cutting-edge, in-depth overview of all the math concepts to achieve 170 in the Quantitative Reasoning portion of the GRE. The book still remains our hallmark. However, the following chapters will give you a lot of tips, tricks, and shortcuts to make your quant preparation more robust and solid.
The term percentage means parts per 100 or “for every hundred”. A fraction whose denominator is 100 is called percentage and the numerator of the fraction is called the rate percent. Thus, when we say a man made a profit of 20 percent we mean to say that he gained Rs.20 for every hundred dollars he invested in the business, i.e., 20/100 dollars for each dollar. The abbreviation of percent is p.c. and it is generally denoted by %.
Let’s start with a number X (= 1 X)X increased by 10% would become X + 0.1 X = 1.1 X
X increased by 1% would become X + 0.01 X = 1.01 X
X increased by 0.1% would become X + 0.001 X = 1.001 X
X decreased by 10% would become X – 0.1 X = 0.9 X
X decreased by 1% would become X – 0.01 X = 0.99 X
X decreased by 0.1% would become X – 0.001 X = 0.999 X
X increased by 200% would become X + 2X = 3X
X decreased by 300% would become X – 3X = –2X
Similarly, you can work mentally with any specifically chosen number (say 500) and work out different answers.
A Percentage can be expressed as a Fraction. 10% can be expressed as \(\frac{10}{100}\) or \(\frac{1}{10}\). To express a percentage as a fraction divide it by 100, \(a\)% \(= \frac{a}{100}\).
To express a fraction as a percent multiply it by 100 ⇒ \(\frac{a}{b}= \left[ \begin{array}{cc|r} (\frac{a}{b}) \times 100 \end{array} \right]\) %
To express percentage as a decimal we remove the symbol % and shift the decimal point by two places to the left. For example, 10% can be expressed as 0.1. 6.5% = 0.065 etc.
To express decimal as a percentage we shift the decimal point by two places to the right and write the number obtained with the symbol % or simply we multiply the decimal with 100. Similarly 0.7 = 70%.
Percent INCREASE and DECREASEIncrease % \(= \frac{Increase}{Original Value} \times 100\)
Decrease % \(= \frac{Decrease }{ Original Value} \times 100
\)
In increase %, the denominator is smaller, whereas, in decrease %, the denominator is larger.
Change % \(= \frac{Change }{ Original Value} \times 100\)
Successive change in percentage. If a number A is increased successively by X% followed by Y%, and then by Z%, then the final value of A will be \(A(1+\frac{X}{100})(1+\frac{Y}{100})(1+\frac{Z}{100})\)
In a similar way, at any point or stage, if the value is decreased by any percentage, then we can replace the same by a negative sign. The same formula can be used for two or more successive changes. The final value of A, in this case, will be \(A(1-\frac{X}{100})(1-\frac{Y}{100})(1-\frac{Z}{100})\)
Also, let us remember that2 = 200% (or 100% increase), 3 = 300% (or 200% increase), 3.26 = 326% (means 226% increase), fourfold (4 times) = 400 % of original = 300% increase, 10 times means 1000% means 900% increase, 0.6 means 60% of the original means 40% decrease, 0.31 times means 31% of the original means 69% decrease etc.
1/2 = 50%, 3/2 = 1 + 1/2 = 100 + 50 = 150%, 5/2 = 2 + 1/2 = 200 + 50 = 250% etc.,
2/3 = 1 - 1/3 = 100 - 33.33 = 66.66%, 4/3 = 1 + 1/3 = 100 + 33.33 = 133.33%,
5/3 = 1 + 2/3 = 100 + 66.66 % = 166.66%, 7/3 = 2 + 1/3 = 200 + 33.33 = 233.33%,
8/3 = 2 + 2/3 = 200 + 66.66 = 3 - 1/3 = 300 - 33.33 = 266.66% etc.
1/4 = 25%, 3/4 = 75%, 5/4 (1 + 1/4) = 125% (= 25 increase), 7/4 (1 + 3/4 = 2 - 1/4) = 175% (= 75% increase), 9/4 (2 + 1/4) = 225% (= 125% increase), 11/4 = 275% = (175% increase).
1/5 = 20%, 2/5 = 40%, 3/5 = 60%, 4/5 = 80%, 6/5 = 120%, 7/5 (1 + 2/5) = 140% etc.
1/6 = 16.66%, 5/6 = 83.33%, 7/6 (1 + 1/6) = 116.66%, 11/6 = 183.33%
1/8 = 12.5%, 3/8 = 37.5%, 5/8 = 62.5%, 7/8 = 87.5%, 9/8 = (1 + 1/8) = 112.5%, 11/8 = (1 + 3/8) = 137.5%, 13/8 = 162.5%, 15/8 = 187.5% etc.
1/9 = 11.11%, 2/9 = 22.22%, 4/9 = 44.44%, 5/9 = 55.55%, 7/9 = 77.77%, 8/9 = 88.88%, 10/9 = 111.11%, 11/9 = (1 + 2/9) = 122.22% etc.
If we have a problem that deals with a discount, it is useful to remember:
Marked Price - Discount = Sale Price. Also Cost Price + Profit = Sale Price.
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