The GRE General Exam - The Definitive Shortcuts Guide ALGEBRA is conceived as the second chapter of the three we will release every Monday in the coming weeks. It is not only just a collection of simple tricks or shortcuts for the quantitative reasoning portion of the GRE exam, but also a compendium if the students want to have, for example, the divisibility rule at a glance without consulting our GRE Math Essentials - A most comprehensive handout!! [COMPLETED] or the GRE - Math Book which I strongly recommend studying for complete and robust quant theory knowledge for the GRE.

The subsequent chapters that will be released are:

• NUMBER SYSTEM Shortcuts
• Geometry Shortcuts

1.
Identities

(a) \((a + b)^2 + (a – b)^2 = 2 (a^2 + b^2)\)

(b) \((a+b)^2-(a-b)^2=4ab\)

(c) \(a^3+b^3=(a+b)(a^2-ab+b^2)\)

(d) \(a^3-b^3=(a-b)(a^2+ab+b^2)\)

(e) \(a^4+a^2+1=(a^2+a+1)(a^2-a+1)\)

(f) If \(a+b+c=0\) then \(a^3+b^3+c^3=3abc\)

(g) \(\frac{(a+b)^2-(a-b)^2}{ab}=4\)

(h) \(\frac{(a+b)^2+(a-b)^2}{a^2+b^2}=2\)

(i) \(a \frac{b}{c}+d \frac{e}{f}+g \frac{h}{i}-j \frac{k}{l}= (a+d+g-j)+\frac{(b}{c}+\frac{ e}{f}+\frac{h}{i}-\frac{k}{l})\)

(l) If \(a+b+c=abc\), then \((\frac{2a}{1-a^2})+(\frac{2b}{1-b^2})+(\frac{2c}{1-c^2})=(\frac{2a}{1-a^2})(\frac{2b}{1-b^2})(\frac{2c}{1-c^2})\)

AND \((\frac{3a-a^3}{1-3a^2})+(\frac{3b-b^3}{1-3b^2})+(\frac{3c-c^3}{1-3c^2})=(\frac{3a-a^3}{1-3a^2})(\frac{3b-b^3}{1-3b^2})(\frac{3c-c^3}{1-3c^2})\)

(m) If \(a_1x+b_1y=c_1\) and \(a_2x+b_2y=c_2\) then

(i) If \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), one solution
(ii) If \(\frac{a_1}{a_2} = \frac{b_1}{b_2}=\frac{c_1}{c_2}\), infinite solutions
(iii) If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\), NO solution

(n) If a and b are roots of \(ax^2+ bx + c = 0\), then \(\frac{1}{a}\) and \(\frac{1}{b}\) are roots of \(cx^2+bx+a=0\)

(o) If a and b are roots of \(ax^2+ bx + c = 0\), then

(i) One root is zero if c = 0.

(ii) Both roots zero if b = 0 and c = 0.

(iii) Roots are reciprocal to each other, if c = a.

(iv) If both roots a and b are positive, then sign of a and b are opposite and sign of c and a are same.

(v) If both roots a and b are negative, then sign of a, b and c are same.


2.
Average

(a) Average of first n natural no. \(= \frac{n+1}{2}\)

(b) Average of first n even no. \(= (n + 1)\)

(c) Average of first n odd no. \(= n\)

(d) Average of sum of square of first n natural no. \(= \frac{(n+ 1 )(2n 1 )}{6}\)

(e) Average of sum of square of first n even no. \(= \frac{2 (n+1)(2n 1)}{3}\)

(f) Average of sum of square of first odd no. \(= (\frac{4n^2-1}{3})\)

(g) If average of some observations is x and a is added in each observation, then new average is \((x + a)\).

(h) If average of some observations is x and a is subtracted in each observation, then new average is \((x – a)\).

(i) If average of some observations is x and each observation multiply by a, then new average is \(ax\).

(l) If average of some observations is x and each observation is divided by a, then new average is \(\frac{x}{a}\)

(m) If average of \(n_1\) is \(A_1\) , & average of \(n_2\) is \(A_2\), then Average of (n_1+n_2) is \(\frac{n_1A_1+n_2A_2}{n_1+n_2}\) and Avergae of (n_1-n_2) is \(\frac{n_1A_1-n_2A_2}{n_1-n_2}\)

(n) When a person is included or excluded the group, then age/weight of that person = No. of persons in group × (Increase /Decrease) in average ± New average.

For example : In a class average age of 15 students is 18 yrs. When the age of teacher is included their average increased by 2 yrs, then find the age of teacher Sol. Age of teacher = 15 × 2 + (18 + 2) = 30 + 20 = 50 yrs.

(o) When two or more than two persons included or excluded the group, then average age of included or excluded person is


\(= \frac{No. of person * (Increase-Decrease) in average ± New average * (No. of person included or excluded)}{No. of included or person}\)

For example : Average weight of 13 students is 44 kg. After including two new students their average weight becomes 48 kg, then find the average weight of two new students. Sol. Average weight of two new students

\(\frac{13*(48-44)+48*2}{2}=\frac{13*44+48*2}{2}=\frac{52+96}{2}=74\) Kg

(p) If a person travels two equal distances at a speed of x km/h and y km/h, then average speed \(= \frac{2xy}{x+y}\) Km/h

(q) If a person travels three equal distances at a speed of x km/h, y km/h and z km/h, then average speed = \(= \frac{3xyz}{xy+yz+zx}\) Km/h



3.
Arithmetic Progression & Geometric Progression

(a) If a, a + d, a + 2d, .... are in A.P., then, nth term of A.P. \(an = a + (n – 1) d\)
Sum of n terms of this \(A.P = Sn = \frac{n}{2} [2a + (n-1) d]\)
Where: a = first term
d = common difference

(b) G.P. = \(a, ar, ar^2\), .........

Then, nth term of G.P. \(a_n = ar^{n-1}\)

Sum of n numbers For \( r>1, Sn = \frac{a (r^n-1)}{r-1}\).

For \(r<1, Sn = \frac{a (1 - r^n)}{ 1-r}\)



3.
Ratio and Proportion

(a) \(\frac{A}{K_1} =\frac{ b}{K_2 }= \frac{c}{K_3} \)….. then \(\frac{(a+b+c+….)}{c }= \frac{(K_1+K_2+K_3+…)}{K_3}\)

For example if \(\frac{P}{3}=\frac{Q}{4}=\frac{R}{7}\) then find \(\frac{P+Q+R}{R}\)

Solution: P=3, Q=4, and R=7 then \(\frac{P+Q+R}{R}=\frac{3+4+7}{7}=2\)


(b) A number added or subtracted from a, b, c & d, so that they are in proportion \(= \frac{(ad-bc)}{(a+d)-(b+c)}\)

For example : When a number should be subtracted from 2, 3, 1 & 5 so that they are in proportion. Find that number.
Sol. Req. No. \(= \frac{2*5-3*1}{(2+5)-(3+1)}=\frac{7}{3}\)


(c) If X part of A is equal to Y part of B, then A : B = Y : X.

For example: If 20% of A = 30% of B, then find A : B.

Sol. \(A : B = \frac{30 }{20 }=\frac{3}{2}=3:2\)


4.
Time, Distance, Work


(a) A can do \(\frac{a}{b}\) part of work in \(t_1\) days and \(\frac{c}{d}\) part of work in \(t_2\) days, then \(\frac{t_1}{\frac{a}{b}}=\frac{t_2}{\frac{c}{d}}\)

(b) If A is K times efficient than B, Then \(T(K + 1) = Kt_B\)
If A is K times efficient than B and takes t days less than B

Then \(T = \dfrac{K_t}{K^2-1}\) or \(\dfrac{t}{K-1} , t_B=\dfrac{t}{K-1}=kt_A\)

(c) If a cistern takes X min to be filled by a pipe but due to a leak, it takes Y extra minutes to be filled, then the time taken by a leak to empty the cistern = \((\frac{X^2+XY}{Y})\) min

If a leak empties a cistern in X hours. A pipe which admits Y liters per hour water into the cistern and now cistern is emptied in Z hours, then the capacity of the cistern is =\( (\frac{X+Y+Z}{Z-X})\) liters

If two pipes A and B fill a cistern in x hours and y hours. A pipe is also an outlet C. If all three pipes are opened together, the tank is full in T hours. Then the time taken by C to empty the full tank is \(= \left[ \begin{array}{cc|r} \frac{xyT}{yT+xT-xy} \end{array} \right]\)


(d) If \(t_1\) and \(t_2\) time taken to travel from A to B and B to A, with speed a km/h and b km/h, then distance from A to B is

\(d=(t_1+t_2) \left( \begin{array}{cc} \dfrac{ab}{a+b} \end{array} \right)\)

\(d=(t_1-t_2) \left( \begin{array}{cc} \dfrac{ab}{a-b} \end{array} \right)\)

\(d=(a-b) \left( \begin{array}{cc} \dfrac{t_1t_2}{t_1-t_2} \end{array} \right)\)


If the 1st part of the distance is covered at the speed of a in \(t_1\) time and the second part is covered at the speed of b in \(t_2\) time, then the average speed \(= \dfrac{at_2+bt_1}{t_1+t_2}\)



5.
Percentage %





For the two articles, the price behaves according to its





If the side of a square or radius of a circle is x% increase/decrease, then its area increase/decrease \(= \left( \begin{array}{cc} 2x \pm \dfrac{x^2}{100} \end{array} \right)\)

If the side of a square, x% increase/decrease then x% its perimeter and diagonal increase/decrease.

If population P increase/decrease at r% rate, then after t years population \(= P \left( \begin{array}{cc} \dfrac{ 100 \pm R}{100} \end{array} \right)^t\)


If population P increase/decrease \(r_1\)% first year, \(r_2\)% increase/decrease second year and \(r_3\)% increase/decrease third year, then after 3 years population \(= P \left( \begin{array}{cc} 1 \pm \dfrac{r_1}{100} \end{array} \right) \left( \begin{array}{cc} 1 \pm \dfrac{r_2}{100} \end{array} \right) \left( \begin{array}{cc} 1 \pm \dfrac{r_3}{100} \end{array} \right)\)



5.
Profit AND Loss


What is Cost Price (C.P.)?
The price at which an article is purchased is called its cost price.

What is Selling Price (S.P.)?
The price at which an article is sold is called its selling price.

What is Profit?
If the selling price of an article is greater than its cost price, we say that there is a profit (or) gain.

What is Loss?
If the selling price of an article is less than its cost price, we say that there is a loss.


If CP of two articles are same, and they sold at



If SP of two articles are same and they sold at



After D% discount, requires P% profit, then total increase in \(C.P.= \left[ \begin{array}{cc|r} \dfrac{P+D}{100-D} \times 100 \end{array} \right]\)


\(M.P. = C.P × \dfrac{100+P}{100-D}\)


Profit %\(= \dfrac{(M.P. -C.P.) \times 100}{C.P.}\)


5.
Simple and Compound Interest


If P = Principal, R = Rate per annum,
T = Time in years, SI = Simple interest,
A = Amount

\(SI=\dfrac{PRT}{100}
\)

\(A = P +SI =P \left[ \begin{array}{cc|r} 1+ \dfrac{RT}{100} \end{array} \right]\)


If P = Principal, A = Amount in n years, R = rate of interest per annum \(A =P \left[ \begin{array}{cc|r} 1+ \dfrac{RT}{100} \end{array} \right]^n\), interest payable annually

\(A =P \left[ \begin{array}{cc|r} 1+ \dfrac{RT}{100} \end{array} \right]^{2n}\) , interest payable half-yearly

\(A =P \left[ \begin{array}{cc|r} 1+ \dfrac{RT}{100} \end{array} \right]^{4n}\) , interest payable quarterly