SSC CHSL (Combined Higher Secondary Level) Quantitative Aptitude Syllabus

What is the full form of BODMAS and the order of operations it dictates?

Brackets, Of, Division, Multiplication, Addition, Subtraction. Solve in that order: brackets first (in order $(), \{\}, []$), then 'of', then $\times$ and $\div$ left to right, then $+$ and $-$.

In BODMAS, how are 'Of', division, and multiplication handled relative to each other?

'Of' is treated as multiplication done before normal division/multiplication. After 'of', division and multiplication have equal priority and are performed left to right.

How do you convert a recurring decimal like $0.\overline{36}$ into a fraction?

Pure recurring: place the repeating block over as many $9$s. $$0.\overline{36} = \frac{36}{99} = \frac{4}{11}.$$

How do you compare two fractions $\frac{a}{b}$ and $\frac{c}{d}$ quickly?

Cross-multiply: compare $a \times d$ with $b \times c$. If $ad > bc$ then $\frac{a}{b} > \frac{c}{d}$ (for positive denominators).

State the formulas for $\sqrt{a} \times \sqrt{b}$ and $\sqrt{\frac{a}{b}}$.

$$\sqrt{a}\times\sqrt{b} = \sqrt{ab}, \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \ (b \neq 0).$$

How is a square root found by the prime factorization method?

Factor the number into primes, pair identical factors, and take one factor from each pair; multiply them. E.g. $\sqrt{144}=\sqrt{2^4\cdot 3^2}=2^2\cdot 3 = 12.$

How do you estimate the number of digits in the square root of a perfect square?

If the number has $n$ digits: square root has $\frac{n}{2}$ digits if $n$ is even, and $\frac{n+1}{2}$ digits if $n$ is odd.

Find the cube root of $1728$ by prime factorization.

$1728 = 2^{6}\times 3^{3}$, so $$\sqrt[3]{1728} = 2^{2}\times 3 = 12.$$

State the basic formula for the average (arithmetic mean) of $n$ numbers.

$$\text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} = \frac{\sum x}{n}.$$

What is the average of the first $n$ natural numbers?

$$\text{Average} = \frac{n+1}{2}.$$ (Since their sum is $\frac{n(n+1)}{2}$.)

What is the average of the first $n$ consecutive odd numbers and of the first $n$ even numbers?

Average of first $n$ odd numbers $= n$. Average of first $n$ even numbers $= n+1$.

For any set of consecutive numbers (arithmetic progression), what is a quick way to find the average?

$$\text{Average} = \frac{\text{First term} + \text{Last term}}{2}.$$

Two groups have averages $A_1$ (with $n_1$ items) and $A_2$ (with $n_2$ items). What is the combined average?

$$\text{Combined average} = \frac{n_1 A_1 + n_2 A_2}{n_1 + n_2}.$$ This is the weighted average.

The average of $n$ numbers is $A$. If each number is increased by $k$, what happens to the average?

The new average becomes $A + k$. Adding a constant $k$ to every term raises the average by $k$ (similarly multiplying every term by $k$ multiplies the average by $k$).

In a replacement problem, the average age of a group changes when one member is replaced. How do you find the new member's value?

New member's value $=$ old member's value $+$ (change in average $\times$ number of members). If average rises by $d$ for $n$ members, the new value is higher by $nd$.

What is the basic formula to convert a quantity to a percentage and to find $x\%$ of $N$?

$$\text{Percentage} = \frac{\text{Value}}{\text{Total}} \times 100, \qquad x\% \text{ of } N = \frac{x}{100} \times N.$$

Convert the fractions $\frac{1}{8}$, $\frac{1}{6}$, and $\frac{1}{3}$ to percentages.

$$\frac{1}{8} = 12.5\%, \quad \frac{1}{6} = 16\tfrac{2}{3}\% \approx 16.67\%, \quad \frac{1}{3} = 33\tfrac{1}{3}\% \approx 33.33\%.$$

If a value increases by $x\%$ then decreases by $x\%$, what is the net percentage change?

There is a net decrease of $$\frac{x^{2}}{100}\%.$$ E.g. $+20\%$ then $-20\%$ gives a $4\%$ net loss.

State the formula for successive (net) percentage change of two changes $a\%$ and $b\%$.

$$\text{Net change} = a + b + \frac{ab}{100}\ \%,$$ where increases are positive and decreases are negative.

A quantity's value increases by $r\%$. By what percentage must it now decrease to return to the original?

$$\text{Required decrease} = \frac{r}{100+r}\times 100\ \%.$$ E.g. for a $25\%$ rise, decrease needed $=20\%$.

If A's income is $r\%$ more than B's, by what percent is B's income less than A's?

$$\frac{r}{100+r}\times 100\ \%.$$ Likewise, if A is $r\%$ less than B, then B is more than A by $\frac{r}{100-r}\times100\%$.

Define a ratio and the role of its terms.

A ratio $a:b$ compares two quantities of the same kind by division, $\frac{a}{b}$. Here $a$ is the antecedent and $b$ is the consequent. Multiplying or dividing both by the same non-zero number leaves the ratio unchanged.

State the property of proportion (the rule of the means and extremes).

If $a:b :: c:d$ then $\frac{a}{b}=\frac{c}{d}$ and $$a \times d = b \times c,$$ i.e. product of extremes $=$ product of means.

What is the mean proportional (geometric mean) between $a$ and $b$?

$$\text{Mean proportional} = \sqrt{ab}.$$ It is the value $x$ such that $a:x :: x:b$.

Define compound ratio and duplicate ratio.

Compound ratio of $a:b$ and $c:d$ is $ac:bd$. Duplicate ratio of $a:b$ is $a^{2}:b^{2}$, and sub-duplicate ratio is $\sqrt{a}:\sqrt{b}$.

In a simple partnership where capitals are invested for the same time, how is profit divided?

Profit is shared in the ratio of the capitals invested. If A, B invest $C_1, C_2$, then $$\text{Profit ratio} = C_1 : C_2.$$

In a partnership with different capitals and different time periods, how is profit divided?

In the ratio of the products (capital $\times$ time): $$C_1 t_1 : C_2 t_2 : \dots$$ This product is the basis of each partner's share.

Define a sleeping (dormant) partner versus a working partner.

A working (active) partner manages the business and may receive a salary/commission besides profit share. A sleeping partner only invests capital and shares profit in ratio of investment without managing.

State the formulas for profit, loss, and their percentages (based on cost price).

$$\text{Profit} = SP - CP, \quad \text{Loss} = CP - SP,$$ $$\text{Profit\%} = \frac{\text{Profit}}{CP}\times100, \quad \text{Loss\%} = \frac{\text{Loss}}{CP}\times100.$$

Express selling price (SP) in terms of cost price (CP) and profit/loss percent.

$$SP = CP\times\frac{100+\text{Profit\%}}{100}, \qquad SP = CP\times\frac{100-\text{Loss\%}}{100}.$$

Define marked price and discount, and give the formula relating them to SP.

Marked price (MP) is the listed/tag price before discount. Discount is a reduction on MP. $$SP = MP\times\frac{100-\text{Discount\%}}{100} = MP - \text{Discount}.$$