Key Tips for Medium Level Percentage Problems
Advance to the next level. Master successive operations, weighted calculation, and discount reversal, essential for more complex problems.
Multiply Variation Factors
When you have successive discounts or increases (e.g., -10% and then +5%), DO
NOT add or subtract them directly. Convert each one to its decimal factor and
then multiply them all.
Example: -10% is $0.90$. +5% is $1.05$. The total factor is $0.90$ times $1.05$.
Use the Weighted Percentage
If you have groups of different sizes (e.g., two batches of products with
different percentages of defects), don't average the percentages. You must
calculate the total part in each group and divide by the total overall.
The size of the group affects the final result.
Divide to Reverse Operations
To find the original price before a discount or increase (e.g., a tax) was
applied, you must divide the final value by the decimal factor. This is key in
problems where the initial amount is missing.
Example: If a price already includes a 10% VAT, divide the final price by
$1.10$.
The Percentage of a Percentage
To calculate the X% of Y% of a quantity, simply multiply the two percentages
converted to decimals. There's no need to calculate the first percentage on the
total and then calculate the second.
Example: 20% of 50% is equivalent to 10% of the total.
Beware of the Comparison Base
A common mistake is confusing the base. If a value A is 20% greater than B, this
does not mean that B is 20% less than A.
The variation formula always divides the difference by the initial value (the
comparison base). Always identify what the base is!
Markup to Nullify a Discount
If you applied a discount (e.g., 20%), the markup you need to apply to the
discounted price to return to the original is always greater than the discount.
This is because the markup is calculated on a smaller base (the discounted
price). To nullify a 20% discount, you need a 25% markup.
Intermediate Level Percentage Exercises
Learn to calculate percentages step by step with intermediate level examples and real-world cases.
Exercise 1 — Successive Discounts on Price
A television has a list price of S/1,500. The store applies two successive discounts: first a 20% for a promotion, and then an 10% additional discount on the already reduced price. What is the final price of the television?
| Initial Price | S/1,500 |
|---|---|
| 1st Discount | 20% |
| 2nd Discount | 10% |
1500 × (1 - 0.20) = 1500 × 0.80 = 1,200
1200 × (1 - 0.10) = 1200 × 0.90 = 1,080
Exercise 2 — Find the Base Price before VAT
An invoice indicates that the final cost of a software, including the Value Added Tax (VAT) of 18%, is S/295. What was the original price of the software before applying the tax?
| Price with VAT | S/295 |
|---|---|
| VAT Rate | 18% |
100% + 18% = 118% (or 1.18)
295 / 1.18 = 250
Exercise 3 — Percentage Variation in Area
A rectangular plot measures 10 m wide and 20 m long. If the width is increased by 10% and the length is reduced by 10%. What is the percentage of area variation of the plot?
| Initial Area | $10 \times 20 = 200 m^2$ |
|---|---|
| Width Change | +10% |
| Length Change | -10% |
Width: 10 × 1.10 = 11 m
Length: 20 × 0.90 = 18 m
Final Area: 11 m × 18 m = 198 m^2
Variation = ((198 - 200) / 200) × 100 = -1%
Exercise 4 — Solution Concentration
There are 20 liters of a solution containing 30% alcohol. If 5 liters of pure alcohol are added to the solution, what will be the new percentage concentration of alcohol in the final solution?
| Initial Volume | 20 liters |
|---|---|
| Initial Concentration | 30% alcohol |
| Alcohol Added | 5 liters |
Initial Alcohol: 20 liters × 0.30 = 6 liters
Final Alcohol: 6 liters + 5 liters = 11 liters
Final Volume: 20 liters + 5 liters = 25 liters
Concentration: (11 liters / 25 liters) × 100 = 44%
Exercise 5 — Best Discount Option
A bicycle costs S/800. Store A offers a single discount of 30%. Store B offers two successive discounts of 20% and then 10%. Which store offers the bicycle at the lowest final price?
| Initial Price | S/800 |
|---|---|
| Option A | 30% (single) |
| Option B | 20% and then 10% |
Price A = 800 × (1 - 0.30) = 800 × 0.70 = 560
Price B = 800 × 0.80 × 0.90 = 800 × 0.72 = 576
S/560 (Store A) vs S/576 (Store B)
Exercise 6 — Simple Annual Compound Interest
A person invests S/5,000 in an account that offers a fixed annual interest of 6%. If the money is not withdrawn, how much money will the investor have after two years? (Assume annual compounding).
| Initial Capital | S/5,000 |
|---|---|
| Annual Rate | 6% |
| Period | 2 years |
1 + 0.06 = 1.06
Final Capital = 5000 × (1.06)²
5000 × 1.1236 = 5,618
Exercise 7 — Population Projection (Total)
In an electoral survey of a city, it was determined that 22,500 people are under 30 years old. If it is known that this group represents exactly 45% of the city's total population, how many inhabitants does the city have in total?
| Part (Under 30) | 22,500 people |
|---|---|
| Percentage Represented | 45% |
Total = (Part / Percentage) × 100
Total = (22500 / 45) × 100
Total = 500 × 100 = 50,000
Exercise 8 — Percentage Profit on Cost
A merchant buys a batch of merchandise for S/4,000 (Cost). If they manage to sell all the merchandise for S/5,800 (Sale), what is the percentage profit they obtained with respect to the cost of the merchandise?
| Initial Cost | S/4,000 |
|---|---|
| Selling Price | S/5,800 |
Profit = Sale - Cost = 5800 - 4000 = 1,800
Percentage = (Profit / Cost) × 100
= (1800 / 4000) × 100 = 45%
Exercise 9 — Percentage Reduction to Reach a Goal
A product is currently sold at S/160 . Management has set a goal to reduce the price to S/148 to be more competitive. What percentage must the discount be to reach this target price?
| Initial Price ($V_i$) | S/160 |
|---|---|
| Target Price ($V_f$) | S/148 |
Discount = Initial - Target = 160 - 148 = 12
Percentage = (Discount / Initial) × 100
= (12 / 160) × 100 = 7.5%
Exercise 10 — Percentage of a Subgroup
In a factory of 800 employees, 60% work in production. Of the production employees, 25% are supervisors. How many employees in the factory are production supervisors ?
| Total Employees | 800 |
|---|---|
| % Production | 60% |
| % Supervisors (of Production) | 25% |
Production = 800 × 0.60 = 480
Supervisors = 480 × 0.25 = 120
Exercise 11 — Find the Base Price with Two Increases
The price of a stock is currently S/330 . This price is the result of the stock rising 10% last month and another 10% this month over last month's value. What was the price of the stock two months ago?
| Final Price | S/330 |
|---|---|
| 1st Increase | +10% (Factor 1.10) |
| 2nd Increase | +10% (Factor 1.10) |
Cumulative Factor = 1.10 × 1.10 = 1.21
Original Price = 330 / 1.21 ≈ 272.73
Exercise 12 — Net Profit Calculation with Taxes
A service was sold for S/4,000 . The total cost to provide the service was S/2,800 . On the profit, the company must pay a tax of 15% . What was the net profit (after taxes)?
| Total Revenue | S/4,000 |
|---|---|
| Total Cost | S/2,800 |
| Tax on Profit | 15% |
Gross Profit = Revenue - Cost = 4000 - 2800 = 1,200
100% - 15% = 85% (or 0.85)
Net Profit = 1200 × 0.85 = 1,020
Exercise 13 — Final Price after Discount and Surcharge
A computer monitor costs S/1,200 . First, a 20% discount is applied for an event, and then the discounted price receives a 10% surcharge for shipping cost. What is the final price of the monitor?
| Initial Price | S/1,200 |
|---|---|
| 1st Operation | -20% (Discount) |
| 2nd Operation | +10% (Surcharge) |
Price with Discount = 1200 × (1 - 0.20) = 1200 × 0.80 = 960
Final Price = 960 × (1 + 0.10) = 960 × 1.10 = 1,056
Exercise 14 — Quantity Needed for a Concentration
A chemist has 15 liters of a solution containing 20% acid. He wants to add pure water (0% acid) so that the new acid concentration is 15% . How many liters of water must he add?
| Initial Volume | 15 liters |
|---|---|
| Initial Concentration | 20% acid |
| Desired Final Concentration | 15% acid |
Acid = 15 liters × 0.20 = 3 liters
Final Total Volume = Acid / Final Concentration = 3 / 0.15 = 20 liters
Water Added = 20 liters - 15 liters = 5 liters
Exercise 15 — Total Population by Subgroup Inference
In an assembly, 35% of attendees voted Yes and 40% voted No. If the remaining 50 people abstained (did not vote), how many attendees were there in total at the assembly?
| % Voted Yes | 35% |
|---|---|
| % Voted No | 40% |
| Abstentions (Part) | 50 people |
% Abstention = 100% - (35% + 40%) = 100% - 75% = 25%
Total = (Part / Percentage) × 100
Total = (50 / 25) × 100 = 2 × 100 = 200
Exercise 16 — Sustained Population Growth
A city has a current population of 120,000 inhabitants . If the annual growth rate remains constant at 3% , what will the city's population be in two years ? (Assume compounded growth).
| Initial Population | 120,000 inhab. |
|---|---|
| Growth Rate | 3% annually |
| Period | 2 years |
1 + 0.03 = 1.03
Final Population = 120000 × (1.03)²
120000 × 1.0609 = 127,308
Exercise 17 — Weighted Percentage in Groups
At a university, the main campus has 2,000 students, of which 40% are foreign. The satellite campus has 500 students, of which 60% are foreign. What is the total percentage of foreign students across both campuses?
| Total Students | 2500 |
|---|---|
| Main Campus (Foreign) | 40% of 2000 |
| Satellite Campus (Foreign) | 60% of 500 |
Main Campus: 2000 × 0.40 = 800
Satellite Campus: 500 × 0.60 = 300
Total Foreign = 800 + 300 = 1,100
Grand Total = 2000 + 500 = 2,500
Percentage = (1100 / 2500) × 100 = 44%
Exercise 18 — Find Price Before Discount and Surcharge
A pair of shoes has a final price of S/144 . It is known that this price resulted from applying a 20% discount to the original price, and then an 8% VAT (tax) on the already discounted price. What was the original price before any discount?
| Final Price | S/144 |
|---|---|
| 1st Operation (Discount) | -20% (Factor 0.80) |
| 2nd Operation (VAT/Surcharge) | +8% (Factor 1.08) |
Discounted Price = 144 / 1.08 = 133.33
Original Price = 133.33 / 0.80 ≈ 166.67
Exercise 19 — Find the Total from the Variation Amount
A company's monthly revenue increased by S/1,200 this month compared to the previous month. If it is known that this increase represents exactly 6% of the previous month's revenue ($V_i$), what was the previous month's revenue?
| Amount of Increase (Part) | S/1,200 |
|---|---|
| % Increase | 6% |
| Total to Find ($V_i$) | Previous revenue |
Total = (Part / Percentage) × 100
Previous Revenue = (1200 / 6) × 100
= 200 × 100 = 20,000
Exercise 20 — Surcharge to Reverse Discount
The original price of a product was discounted by 20% . What surcharge percentage must be applied to the discounted price so that the final price is exactly equal to the original price?
| Initial Discount | 20% (Factor 0.80) |
|---|---|
| Initial Price (Total) | Assume 100 (or 1) |
| Discounted Price | 80 (or 0.8) |
Original Price - Discounted Price = 100 - 80 = 20 (or $1 - 0.8 = 0.2$)
Surcharge % = (Difference / Discounted Price) × 100
= (20 / 80) × 100 = 0.25 × 100 = 25%