Finding a percentage of a number is one of the most fundamental numerical operations. To find X% of Y, multiply Y by X and divide by 100, or equivalently, multiply Y by the decimal form of X (X/100). This skill is essential for calculating discounts, tips, taxes, and portions.

X% of Y = Y × (X/100) = Y × 0.0X

What is 25% of 80? → 80 × 0.25 = 20
What is 15% of 200? → 200 × 0.15 = 30
What is 40% of 150? → 150 × 0.40 = 60

How to recognize it

Look for 'what is X% of Y' phrasing
Convert the percentage to decimal (divide by 100)
Multiply the base number by the decimal

Common mistakes

Forgetting to divide the percentage by 100
Multiplying by the percentage directly (25 instead of 0.25)
Confusing 'of' with 'off' (discount calculations)

Step-by-step walkthrough

What is 15% of 80?

Convert percentage to decimal: 15/100 = 0.15
Multiply the base number by the decimal: 80 x 0.15
Calculate: 80 x 0.15 = 12

Answer: 12

Learn 2

Reverse Percentage

Find the original value before a percentage change was applied.

Reverse percentage problems ask you to work backwards from a final value to find the original. If something increased by X%, the final value equals the original times (1 + X/100). To find the original, divide the final value by this multiplier. This is trickier than forward percentages because you can't simply subtract the percentage.

If Final = Original × (1 + X/100), then Original = Final ÷ (1 + X/100)

After a 20% increase, a price is $120. Original = $120 ÷ 1.20 = $100
After a 25% discount, a price is $60. Original = $60 ÷ 0.75 = $80
After a 10% raise, salary is $55,000. Original = $55,000 ÷ 1.10 = $50,000

How to recognize it

You're given the final value after a percentage change
You need to find what the value was before the change
Keywords: 'original price', 'was before', 'started at'

Common mistakes

Subtracting the percentage from the final (e.g., $120 - 20% = $96, wrong!)
Using the wrong multiplier (1.20 for increase, 0.80 for 20% decrease)
Confusing increase (multiply by >1) with decrease (multiply by <1)

Step-by-step walkthrough

After a 25% increase, a price is $150. What was the original price?

Identify the multiplier for a 25% increase: 1 + 25/100 = 1.25
Set up the equation: Original x 1.25 = $150
Divide the final value by the multiplier: $150 / 1.25
Calculate: $150 / 1.25 = $120

Answer: $120

Learn 3

Percentage Change

Calculate the result after increasing or decreasing a value by a percentage.

Percentage change problems involve applying an increase or decrease to a starting value. For an increase of X%, multiply by (1 + X/100). For a decrease of X%, multiply by (1 - X/100). This is more direct than reverse percentage since you're moving forward from the original to the final value.

Increase: Final = Original × (1 + X/100). Decrease: Final = Original × (1 - X/100)

100 increased by 20% = 100 × 1.20 = 120
80 decreased by 25% = 80 × 0.75 = 60
150 increased by 10% = 150 × 1.10 = 165

How to recognize it

Given a starting value and a percentage change
Asked for the new/final value
Keywords: 'increases by', 'decreases by', 'goes up', 'goes down'

Common mistakes

Adding/subtracting the percentage instead of using a multiplier
Confusing 'increased by 20%' with 'increased to 120%'
Using the wrong direction (1.20 for increase, 0.80 for 20% decrease)

Step-by-step walkthrough

A shirt costs $60. It goes on sale for 30% off. What is the sale price?

Identify this as a percentage decrease: 30% off
Calculate the multiplier for a 30% decrease: 1 - 30/100 = 0.70
Multiply the original price by the multiplier: $60 x 0.70
Calculate: $60 x 0.70 = $42

Answer: $42

Learn 4

Simple Ratio

Find unknown values when two quantities are in a given ratio.

Ratio problems express a relationship between two quantities. If A:B = 2:3 and you know A=10, you can find B by first finding the multiplier (10÷2=5) and then applying it to the other part (3×5=15). The ratio tells you the relative sizes, and the multiplier scales it to actual values.

If A:B = m:n and A = x, then B = x × (n/m)

If A:B = 2:3 and A = 10, then B = 10 × (3/2) = 15
If X:Y = 3:4 and Y = 20, then X = 20 × (3/4) = 15
If P:Q = 5:2 and P = 35, then Q = 35 × (2/5) = 14

How to recognize it

Two quantities described as 'in the ratio' of X:Y
One value is given, the other is unknown
Find the multiplier by dividing the known value by its ratio part

Common mistakes

Dividing by the wrong ratio part
Confusing which value corresponds to which part of the ratio
Adding/subtracting instead of multiplying

Step-by-step walkthrough

The ratio of boys to girls in a class is 3:5. If there are 12 boys, how many girls are there?

Identify the ratio parts: boys = 3, girls = 5
Find the multiplier: 12 boys / 3 parts = 4
Apply the multiplier to find girls: 5 parts x 4
Calculate: 5 x 4 = 20 girls

Answer: 20 girls

Learn 5

Ratio Division

Divide a total amount according to a given ratio.

Ratio division problems ask you to split a total into parts according to a ratio. First, add the ratio parts to find the total number of 'shares'. Then divide the total amount by this number to find the value of one share. Finally, multiply by each ratio part to get the actual amounts.

Total T split in ratio a:b → Part A = T × a/(a+b), Part B = T × b/(a+b)

Split $100 in ratio 3:2 → Shares = 5, each share = $20 → Parts are $60 and $40
Divide 45 items in ratio 4:5 → Shares = 9, each = 5 → Parts are 20 and 25
Split 120 in ratio 1:2:3 → Shares = 6, each = 20 → Parts are 20, 40, and 60

How to recognize it

A total amount needs to be divided
The division follows a specified ratio
Add ratio parts first to find total shares

Common mistakes

Forgetting to add ratio parts to find total shares
Dividing by only one ratio part instead of the sum
Mixing up which part corresponds to which share

Step-by-step walkthrough

Divide $180 between two people in the ratio 2:7.

Add the ratio parts to find total shares: 2 + 7 = 9
Find the value of one share: $180 / 9 = $20
Calculate the first person's share: 2 x $20 = $40
Calculate the second person's share: 7 x $20 = $140

Answer: $40 and $140

Learn 6

Work Rate Problems

Calculate time or workers needed when the total work stays constant.

Work rate problems are based on the formula: Work = Rate × Time, or equivalently, Work = Workers × Days. If the total work is constant, increasing workers decreases time proportionally. The key insight is that 'worker-days' (or 'machine-hours') represents the total work, and this stays the same regardless of how many workers do the job.

Total Work = Workers × Time. If W1 × T1 = W2 × T2, then T2 = (W1 × T1) / W2

4 workers in 12 days = 48 worker-days. 6 workers = 48 ÷ 6 = 8 days
3 machines in 10 hours = 30 machine-hours. 5 machines = 30 ÷ 5 = 6 hours
2 people in 15 days = 30 person-days. 5 people = 30 ÷ 5 = 6 days

How to recognize it

Problem mentions workers/machines and time to complete a job
Asks how long with different number of workers
Total work (job size) remains constant

Common mistakes

Thinking more workers means more time (it's inverse)
Adding workers instead of multiplying
Forgetting that work = workers × time

Step-by-step walkthrough

5 workers can complete a project in 12 days. How many days would it take 10 workers?

Calculate total work in worker-days: 5 workers x 12 days = 60 worker-days
The total work stays constant at 60 worker-days
Divide total work by new number of workers: 60 / 10 workers
Calculate: 60 / 10 = 6 days

Answer: 6 days

Learn 7

Profit Calculation

Calculate profit, cost price, or selling price from given values.

Profit calculations involve three key values: cost price (what you paid), selling price (what you sold for), and profit (the difference). Profit = Selling Price - Cost Price. Profit margin as a percentage is calculated relative to the cost price: Profit % = (Profit / Cost) × 100.

Profit = Selling Price - Cost Price. Profit % = (Profit / Cost) × 100

Buy at $80, sell at $100 → Profit = $20, Margin = 20/80 × 100 = 25%
Cost $50, 40% markup → Selling = $50 × 1.40 = $70, Profit = $20
Sold at $90 with $18 profit → Cost = $90 - $18 = $72

How to recognize it

Problem mentions cost, selling price, or profit
May ask for profit amount or profit percentage
Markup problems are percentage-based profit calculations

Common mistakes

Confusing profit margin with markup
Calculating percentage based on selling price instead of cost
Mixing up which value to subtract from which

Step-by-step walkthrough

A shop buys an item for $60 and sells it for $75. What is the profit margin percentage?

Calculate the profit: Selling Price - Cost Price = $75 - $60 = $15
Divide profit by cost price: $15 / $60
Convert to percentage: ($15 / $60) x 100
Calculate: 0.25 x 100 = 25%

Answer: 25%

Learn 8

Compound Percentage

Calculate the result of applying multiple percentage changes in sequence.

Compound percentage problems involve applying multiple percentage changes one after another. Each change is applied to the result of the previous change, not the original value. A 10% increase followed by a 10% increase is NOT a 20% total increase—it's 1.10 × 1.10 = 1.21, or 21%.

After changes of X% then Y%: Final = Original × (1 + X/100) × (1 + Y/100)

100 increased by 10% then 20%: 100 × 1.10 × 1.20 = 132 (not 130)
200 up 25% then down 20%: 200 × 1.25 × 0.80 = 200 (back to start)
Price up 50% then down 50%: Not zero! 100 × 1.50 × 0.50 = 75

How to recognize it

Multiple percentage changes applied in sequence
Each change applies to the result of the previous
Keywords: 'then', 'followed by', 'successive'

Common mistakes

Adding percentages together (10% + 20% = 30%, wrong!)
Thinking opposite changes cancel out (up 50%, down 50% ≠ original)
Applying both percentages to the original value

Step-by-step walkthrough

A price of $200 increases by 20%, then decreases by 10%. What is the final price?

Calculate the multiplier for 20% increase: 1 + 0.20 = 1.20
Calculate the multiplier for 10% decrease: 1 - 0.10 = 0.90
Apply both multipliers in sequence: $200 x 1.20 x 0.90
Calculate step by step: $200 x 1.20 = $240, then $240 x 0.90 = $216

Answer: $216

Learn 9

Combined Rates

Calculate how long tasks take when multiple workers or machines work together.

When multiple workers or machines work together, their rates add up. If Machine A produces 5 units/hour and Machine B produces 3 units/hour, together they produce 8 units/hour. To find time for a task, divide the total work by the combined rate. This also applies to pipes filling tanks or people completing tasks.

Combined Rate = Rate A + Rate B. Time = Total Work / Combined Rate

A makes 4/hr, B makes 6/hr → Together 10/hr → 50 units take 5 hours
Pipe A fills in 6 hrs, B in 3 hrs → Rates: 1/6 + 1/3 = 1/2 → Together: 2 hours
Worker A: 8 items/day, B: 12 items/day → 100 items = 100/20 = 5 days together

How to recognize it

Multiple workers/machines working simultaneously
Individual rates are given (units per time)
Asked for time when working together

Common mistakes

Averaging rates instead of adding them
Adding times instead of rates (if A takes 2hr and B takes 3hr, together is NOT 5hr)
Forgetting to convert to consistent rate units

Step-by-step walkthrough

Pipe A can fill a tank in 4 hours. Pipe B can fill the same tank in 6 hours. How long to fill the tank using both pipes?

Calculate Pipe A's rate: 1 tank / 4 hours = 1/4 tank per hour
Calculate Pipe B's rate: 1 tank / 6 hours = 1/6 tank per hour
Add the rates: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 tank per hour
Calculate time: 1 tank / (5/12 per hour) = 12/5 = 2.4 hours

Answer: 2.4 hours (or 2 hours 24 minutes)

Learn 10

Average with a Missing Value

Given a target mean and all but one data value, solve for the missing value.

If the arithmetic mean of n values must equal M, then the total sum of the values must equal n · M. To find a missing value, compute the required total (n · M) and subtract the sum of the known values: missing = n · M − Σ(known). This technique applies whenever one item is unknown — the next test score, the final month's sales, a missing measurement — and a target average is specified.

missing = n · M − Σ(known values)

Four test scores average 80. Three scores are 75, 82, 78. Missing = 4·80 − (75+82+78) = 320 − 235 = 85
Five months' sales average $10,000. Four months total $38,500. Missing = 5·10000 − 38500 = $11,500
Six measurements average 12.5. Five sum to 70. Missing = 6·12.5 − 70 = 5

How to recognize it

A mean is given but one value is unknown
Keywords: 'target average', 'needs to average', 'what score is required'
The total count of values n is known

Common mistakes

Forgetting to multiply the mean by n to get the required total
Subtracting in the wrong order (known − total instead of total − known)
Using n − 1 instead of n when counting the total number of values

Learn 11

Combinations

The number of ways to choose k items from n without regard to order.

The number of combinations C(n, k) (also written nCk or 'n choose k') counts the unordered selections of k items from a set of n distinct items. It equals the binomial coefficient n! / (k! · (n − k)!). Combinations differ from permutations (where order matters): C(n, k) = P(n, k) / k!. Symmetry: C(n, k) = C(n, n − k), so choosing k to include is equivalent to choosing n − k to exclude.

C(n, k) = n! / (k! · (n − k)!)

C(5, 2) = 10 (ten ways to pick 2 from 5, such as {1,2}, {1,3}, ..., {4,5})
C(10, 3) = 120
C(52, 5) = 2,598,960 (five-card poker hands)

How to recognize it

The problem asks 'how many ways' without regard to order
Keywords: 'choose', 'select', 'committee', 'team', 'hand of cards'
If rearranging the chosen items yields the same selection, use combinations, not permutations

Common mistakes

Using permutations when order does not matter
Forgetting the k! in the denominator
Confusing C(n, k) with n · k or with n^k

Learn 12

Compound Interest

Interest computed on the principal and on previously accumulated interest, growing geometrically.

Compound interest adds interest to the principal at the end of each compounding period so that the next period earns interest on the larger amount. With principal P, annual interest rate r, compounding n times per year, and time t in years, the final amount is A = P · (1 + r/n)^(n·t). For annual compounding (n = 1) this reduces to A = P · (1 + r)^t. Compound interest grows geometrically (multiplicatively), unlike simple interest which grows linearly.

A = P · (1 + r/n)^(n·t)

$1,000 at 5% annually for 3 years → A = 1000 · 1.05^3 ≈ $1,157.63
€500 at 4% compounded quarterly for 2 years → A = 500 · (1.01)^8 ≈ €541.43
$2,000 at 6% compounded monthly for 1 year → A = 2000 · (1.005)^12 ≈ $2,123.36

How to recognize it

Problem mentions 'compound', 'compounded monthly/yearly/daily', or 'annual yield'
Growth multiplies each period rather than adds
The balance accelerates over time (convex upward curve)

Common mistakes

Using simple interest when compounding is specified
Forgetting to divide the rate by the compounding frequency (r/n, not r)
Using n in place of n·t in the exponent

Learn 13

Currency Conversion

Converting an amount from one currency to another by multiplying by the exchange rate.

An exchange rate expresses how much of currency B you get per unit of currency A. To convert an amount X from currency A to currency B, multiply by the rate: X_B = X_A · rate(A→B). The reverse rate is 1 / rate(A→B). When converting between two currencies that aren't directly quoted, use a cross-rate through a base currency (often USD or EUR): rate(A→C) = rate(A→B) · rate(B→C).

X_B = X_A · rate(A→B); rate(B→A) = 1 / rate(A→B)

Convert 100 USD to EUR at 1 USD = 0.85 EUR → 100 · 0.85 = 85 EUR
Convert 500 EUR to USD at 1 USD = 0.85 EUR → 500 / 0.85 ≈ 588.24 USD
USD → GBP via EUR: 1 USD = 0.85 EUR, 1 EUR = 0.86 GBP → 100 USD = 100 · 0.85 · 0.86 = 73 GBP

How to recognize it

Problem mentions currencies, exchange rates, or conversions
A rate like '1 USD = 0.85 EUR' is given
Questions may require reversing a rate or chaining through a base currency

Common mistakes

Reversing the direction of the rate (using rate(A→B) when B→A is needed)
Multiplying when you should divide or vice versa
Rounding too aggressively before the final answer

Learn 14

Mixture Problem

Compute the concentration of a combined mixture from the concentrations and amounts of its components.

When two solutions with volumes V_1 and V_2 and concentrations c_1 and c_2 are combined, the resulting mixture has volume V_1 + V_2 and concentration c_final = (c_1 · V_1 + c_2 · V_2) / (V_1 + V_2). This is simply a weighted average of the input concentrations, with each concentration weighted by its volume. The same logic extends to three or more components: multiply each component's concentration by its amount, sum, and divide by the total amount.

c_final = Σ(c_i · V_i) / Σ V_i

Mix 2 L of 10% saline with 3 L of 20% saline → c = (0.10·2 + 0.20·3)/5 = 0.16 = 16%
Blend 500 g of 40% silver alloy with 300 g of 60% alloy → c = (0.40·500 + 0.60·300)/800 = 47.5%
Dilute 1 L of 25% sugar solution with 3 L of water (0%) → c = (0.25·1 + 0·3)/4 = 6.25%

How to recognize it

Two or more amounts with different concentrations are being combined
Keywords: 'mix', 'blend', 'dilute', 'alloy', 'solution'
The answer is a combined concentration or a required amount to reach a target

Common mistakes

Averaging concentrations directly (without weighting by volume)
Forgetting that adding pure solvent is a component with concentration 0
Mixing units (mL with L, g with kg)

Learn 15

Probability

The likelihood of an event, expressed as a number between 0 (impossible) and 1 (certain).

For a finite sample space of equally likely outcomes, the probability of an event E is the ratio of favorable outcomes to total outcomes: P(E) = |E| / |S|. Probabilities satisfy Kolmogorov's axioms: 0 ≤ P(E) ≤ 1, P(sample space) = 1, and for mutually exclusive events P(A ∪ B) = P(A) + P(B). For independent events, P(A and B) = P(A) · P(B). Complement rule: P(not E) = 1 − P(E).

P(E) = favorable outcomes / total outcomes; 0 ≤ P(E) ≤ 1

Rolling a 6 on a fair die → P = 1/6
Drawing an ace from a 52-card deck → P = 4/52 = 1/13
Two independent coin flips both heads → P = 1/2 · 1/2 = 1/4

How to recognize it

The problem asks 'what is the chance' or 'what is the probability'
A sample space of equally likely outcomes is described (dice, cards, balls in an urn)
Compound events combine with 'and' (multiply if independent) or 'or' (add if disjoint)

Common mistakes

Adding probabilities of events that are not mutually exclusive without subtracting overlap
Multiplying probabilities of events that are not actually independent
Forgetting the complement rule when the 'at least one' form is easier

Learn 16

Simple Interest

Interest computed only on the original principal, growing linearly with time.

Simple interest is computed as a fixed rate of the original principal for each period, with no compounding. If P is the principal, r is the annual interest rate (expressed as a decimal), and t is the time in years, then the interest earned is I = P · r · t and the final amount is A = P · (1 + r · t). Simple interest grows linearly with time — every year adds the same amount of interest.

I = P · r · t; A = P · (1 + r · t)

$1,000 at 5% for 3 years → I = 1000 · 0.05 · 3 = $150; A = $1,150
€500 at 4% for 2 years → I = 500 · 0.04 · 2 = €40; A = €540
£2,000 at 6% for 6 months (0.5 years) → I = 2000 · 0.06 · 0.5 = £60

How to recognize it

The problem explicitly states 'simple interest'
Interest is the same each period (no compounding language)
Rate is quoted per time unit and you multiply by elapsed time

Common mistakes

Using the compound interest formula when simple interest is intended
Forgetting to convert percentage rates to decimals (5% = 0.05)
Mixing time units (rate annual, time in months without converting)

Learn 17

Speed, Distance, and Time

Using the relationship speed = distance ÷ time to solve for any of the three quantities.

For motion at constant speed, the three quantities distance d, speed v, and time t are linked by d = v · t, from which v = d / t and t = d / v. When speed changes over different legs of a trip, the overall average speed is total distance divided by total time (not a plain average of the individual speeds). Units must be consistent across all three quantities — if distance is in kilometres and time in hours, speed is in km/h.

d = v · t; v = d / t; t = d / v; average speed = total distance / total time

A car travelling 60 km in 1.5 h has speed 40 km/h
Walking 5 km at 4 km/h takes 1.25 h (1 h 15 min)
Leg 1: 30 km at 60 km/h (0.5 h); Leg 2: 30 km at 30 km/h (1.0 h) → average = 60 km / 1.5 h = 40 km/h

How to recognize it

The problem mentions distance, speed, time, or any two of them
You are asked to find the unknown third quantity
Multi-leg trips require total distance and total time for the average speed

Common mistakes

Averaging speeds directly across different-length legs
Mixing units (km + miles, hours + minutes)
Computing v = t / d instead of d / t

Learn 18

Weighted Average

An average in which each value contributes according to its weight, not equally.

The weighted average of values x_1, x_2, ..., x_n with weights w_1, w_2, ..., w_n is defined as x̄ = Σ(w_i · x_i) / Σ w_i. When all weights are equal, the weighted average reduces to the ordinary arithmetic mean. Weighted averages are the correct tool whenever values carry different importance, sample sizes, or frequencies — for example, combining test scores from classes of different sizes, or computing a GPA where each course counts for a given number of credits.

x̄ = Σ(w_i · x_i) / Σ w_i

Test scores: class A avg 80 (20 students), class B avg 70 (30 students) → weighted avg = (20·80 + 30·70) / 50 = 74
GPA: A (3 credits), B (4 credits), C (2 credits) → weighted avg uses credits as weights
Portfolio return: investments weighted by dollar amount invested

How to recognize it

Values carry different sample sizes, importance, or frequencies
Keywords: 'weighted', 'by credits', 'weighted by', 'per unit'
Equal weights → ordinary mean; unequal weights → weighted mean

Common mistakes

Taking a plain mean when values have different weights
Forgetting to divide by the sum of weights
Using counts and values in the wrong positions

Worked examples

The worked examples below preserve the original practice set at every difficulty. Open an example to compare the prompt, answer choices, correct answer, and explanation.

Example

Explanation

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