Numerical Ability formula sheet for CUET's General Aptitude Test
Every formula and shortcut for the twelve Numerical Ability chapters on one page: numbers, percentages, ratios, averages, interest, work, speed and ages.
This is the page to read the night before the test. Each section is the short version of a chapter; the chapter notes have the worked examples, and every chapter has its own practice questions.
What divisibility rules should you know by heart?
A number is divisible by 2 if its last digit is even, by 3 if its digit sum is divisible by 3, by 4 if its last two digits are, by 5 if it ends in 0 or 5, by 8 if its last three digits are, by 9 if its digit sum is, and by 11 if the difference between the sums of alternate digits is 0 or a multiple of 11.
| Divisor | Test |
|---|---|
| 6 | divisible by 2 and by 3 |
| 12 | divisible by 3 and by 4 |
| 15 | divisible by 3 and by 5 |
| 7, 11, 13 | 1001 = 7 × 11 × 13, so a six-digit number of the form abcabc is divisible by all three |
Other number facts that save time:
- Sum of the first n natural numbers = n(n + 1) ÷ 2.
- Sum of the first n odd numbers = n².
- Sum of the squares of the first n natural numbers = n(n + 1)(2n + 1) ÷ 6.
- Unit digit of a power: the unit digits of 2, 3, 7 and 8 repeat every 4 powers; 4 and 9 repeat every 2; 0, 1, 5 and 6 never change.
- Number of factors of N = p^a × q^b × r^c is (a + 1)(b + 1)(c + 1).
- Dividend = Divisor × Quotient + Remainder.
See the number system chapter.
How are HCF and LCM related?
For two numbers, HCF × LCM = the product of the numbers. The HCF always divides the LCM.
- HCF of fractions = HCF of numerators ÷ LCM of denominators.
- LCM of fractions = LCM of numerators ÷ HCF of denominators.
- Largest number that divides a, b and c leaving the same remainder = HCF of (a − b), (b − c) and (a − c).
- Smallest number that leaves remainder r when divided by each of x, y and z = LCM(x, y, z) + r.
- Bells, lights or runners that start together meet again after LCM of their intervals.
- Largest tile or container size that fits exactly = HCF of the lengths or volumes.
See HCF and LCM.
What order do you simplify an expression in?
Brackets first, then "of", then division and multiplication from left to right, then addition and subtraction from left to right. "Of" means multiply, and it is done before a plain ÷ sign.
Algebraic identities that turn long arithmetic into one line:
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- a² − b² = (a + b)(a − b)
- a³ + b³ = (a + b)(a² − ab + b²)
- a³ − b³ = (a − b)(a² + ab + b²)
For approximation, round each number to one it is easy to work with, and pick the option nearest to your estimate; the options are usually far enough apart. See simplification and approximation.
What are the percentage formulas you need?
Percentage change = (Change ÷ Original value) × 100. The base is always the value you start from.
| Situation | Formula |
|---|---|
| x% of y | x × y ÷ 100 (and x% of y = y% of x) |
| Two successive changes of a% and b% | net change = a + b + (a × b ÷ 100) %, with a fall written as negative |
| Rise of r% undone by a fall of | r ÷ (100 + r) × 100 % |
| Price rises r%, spending unchanged: cut in use | r ÷ (100 + r) × 100 % |
| Population P growing r% a year for n years | P × (1 + r ÷ 100)ⁿ |
| A is r% more than B, so B is less than A by | r ÷ (100 + r) × 100 % |
Fraction–percentage pairs worth knowing: ½ = 50%, ⅓ = 33⅓%, ¼ = 25%, ⅕ = 20%, ⅙ = 16⅔%, ⅛ = 12.5%, 1/9 = 11 1/9 %, 1/11 = 9 1/11 %, 1/12 = 8⅓%. See percentages.
How do ratio and proportion formulas work?
A ratio a : b compares two quantities of the same kind; multiplying or dividing both terms by the same number does not change it. To share an amount S in the ratio a : b, the parts are S × a ÷ (a + b) and S × b ÷ (a + b).
- If a : b and b : c are given, make the common term equal, then read off a : b : c.
- In a proportion a : b = c : d, the product of extremes equals the product of means: a × d = b × c.
- Mean proportional of a and b = √(ab). Third proportional to a and b = b² ÷ a. Fourth proportional to a, b and c = bc ÷ a.
- Compounded ratio of a : b and c : d = ac : bd.
- Direct proportion: x ÷ y stays constant. Inverse proportion: x × y stays constant.
See ratio and proportion.
What are the average formulas?
Average = Sum of values ÷ Number of values, so Sum = Average × Number. Almost every average question is solved by turning averages back into sums.
- Weighted average of groups = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).
- Average of consecutive numbers, or of any arithmetic progression, = (first + last) ÷ 2.
- A new value joins n values and the average rises by d: the new value = old average + (n + 1) × d.
- One value is replaced and the average of n values changes by d: new value − old value = n × d.
- Average speed over two equal distances at speeds x and y = 2xy ÷ (x + y), never (x + y) ÷ 2.
See averages.
How does the alligation rule work?
When two ingredients priced c₁ (cheaper) and c₂ (dearer) are mixed to get a mean price m, the quantities are in the ratio (c₂ − m) : (m − c₁). The same rule works for concentrations, averages, speeds and profit percentages.
cheaper c₁ dearer c₂
\ /
mean price m
/ \
c₂ − m m − c₁
(cheaper) (dearer)
Repeated replacement: from a container holding x litres of pure liquid, y litres are taken out and replaced with water n times. The pure liquid left = x × (1 − y ÷ x)ⁿ. See mixtures and alligation.
What are the profit, loss and discount formulas?
Profit or loss is always a percentage of the cost price; discount is always a percentage of the marked price.
| Quantity | Formula |
|---|---|
| Profit % | (SP − CP) ÷ CP × 100 |
| Loss % | (CP − SP) ÷ CP × 100 |
| SP at a profit of p% | CP × (100 + p) ÷ 100 |
| SP at a loss of l% | CP × (100 − l) ÷ 100 |
| Discount % | (MP − SP) ÷ MP × 100 |
| Successive discounts a% and b% | single discount = a + b − (a × b ÷ 100) % |
| Buy x, get y free | discount on the goods received = y ÷ (x + y) × 100 % |
| Marked up m%, then discount d% | profit % = m − d − (m × d ÷ 100) |
Two articles sold at the same price, one at a gain of x% and one at a loss of x%, always make an overall loss of x² ÷ 100 %. A dishonest dealer who gives g grams for every 1,000 grams charged gains (1000 − g) ÷ g × 100 %. See profit, loss and discount.
What are the simple and compound interest formulas?
Simple interest SI = P × R × T ÷ 100, and the amount A = P + SI. Compound interest makes the amount A = P × (1 + R ÷ 100)ᵀ, and CI = A − P.
- Compounded half-yearly: use R ÷ 2 for the rate and 2T for the number of periods. Quarterly: R ÷ 4 and 4T.
- Difference between CI and SI on P at R% for 2 years = P × (R ÷ 100)².
- For 3 years, the difference = P × (R ÷ 100)² × (3 + R ÷ 100).
- For 1 year compounded annually, CI and SI are equal.
- A sum doubles in T years at simple interest when R × T = 100.
- At compound interest, a sum that doubles in n years becomes 4 times in 2n years and 8 times in 3n years.
See simple and compound interest.
How do you solve time and work questions quickly?
If A finishes a job in a days, A does 1 ÷ a of it each day. Rates add when people work together: A and B together finish in ab ÷ (a + b) days.
- M₁ × D₁ × H₁ ÷ W₁ = M₂ × D₂ × H₂ ÷ W₂ (men, days, hours per day, work).
- Efficiency is the inverse of time: if A is twice as efficient as B, A takes half the time.
- Wages are shared in the ratio of work done, which for the same number of days is the ratio of efficiencies.
- The LCM method: take the total work as the LCM of the individual times, so every rate is a whole number of units a day.
- Pipes: a filling pipe's rate is positive, a leak or an emptying pipe's rate is negative. A pipe that fills in x hours together with one that empties in y hours (y greater than x) fills the tank in xy ÷ (y − x) hours.
See time and work.
What are the speed, time and distance formulas?
Distance = Speed × Time. To change km/h to m/s multiply by 5 ÷ 18; to change m/s to km/h multiply by 18 ÷ 5.
| Situation | Formula |
|---|---|
| Relative speed, same direction | difference of the speeds |
| Relative speed, opposite directions | sum of the speeds |
| Train passes a pole or a person standing still | time = train length ÷ train speed |
| Train passes a platform or a bridge | time = (train length + platform length) ÷ train speed |
| Two trains pass each other | time = (sum of lengths) ÷ relative speed |
| Downstream speed | boat speed + stream speed |
| Upstream speed | boat speed − stream speed |
| Speed in still water | (downstream + upstream) ÷ 2 |
| Speed of the stream | (downstream − upstream) ÷ 2 |
If the speed ratio is a : b over the same distance, the time ratio is b : a. See time, speed and distance.
How do you set up age and number problems?
Name one unknown, usually the youngest person's present age or the smaller number, as x, and write every sentence of the question as an equation in x. The difference between two people's ages never changes over time.
- "n years ago" means subtract n from every present age; "after n years" means add n to every age.
- If the ratio of two ages is a : b now, write the ages as ax and bx.
- A two-digit number with tens digit x and units digit y is 10x + y; with the digits reversed it is 10y + x. The two differ by 9 × (x − y).
- Consecutive integers: x, x + 1, x + 2. Consecutive odd or even numbers: x, x + 2, x + 4.
- Two numbers with sum S and difference D are (S + D) ÷ 2 and (S − D) ÷ 2.
See ages and problems on numbers.
Which traps cost the most marks in Numerical Ability?
Most lost marks come from using the wrong base or the wrong unit, not from hard arithmetic.
- Percentage change divided by the new value instead of the old.
- Discount taken on the cost price instead of the marked price.
- Average speed taken as the plain average of two speeds.
- km/h and m/s mixed in one formula.
- The train's own length forgotten when it crosses a platform.
- Successive changes added straight up (20% then 10% is 32%, not 30%).
- Half-yearly interest worked at the yearly rate.
- An age ratio "after n years" applied to present ages.
- "Of" done after a division that comes before it.
- An approximation rounded so hard that two options fit.