Essential Tips for Difficult Level Percentages
Master complex logic: use equations for unknown values, apply percentages to dimensions (area/volume), and solve increments and decrements of three or more steps.
Define the Unknown with Algebra
When the problem asks you to find the initial value or an intermediate
percentage , establish the unknown quantity as the variable $X$ .
Write the complete chain of operations (variation factors) equaling the final
result. This transforms complex narrative problems into a simple equation.
Percentage Change in Area/Volume
If the dimensions of a figure (side, radius, edge) change by a percentage, the
area and volume change quadratically or cubically.
Example: A 10% increase in the side of a square is not a 10% increase in area.
Use the variation factor twice for the area ($1.10 \times 1.10 = 1.21$, a 21%
increase).
Set a Reference Base of 100
In population or mixture problems where only percentages are given (no absolute
numbers), assume the Initial Total is 100 (or 1000).
This allows you to work with simple absolute numbers to find the final
proportions. Once the proportion is found, you can scale it to the real total if
necessary.
Relative Percentage Difference
Distinguish between an absolute percentage change and a relative change .
Example: If efficiency goes from 60% to 90%, the absolute increase is 30%. But
the *relative* increase (or yield) is $30/60 = 0.5$, meaning a 50% increase over
the initial base.
Use Fractions in Nested Calculations
When you have a series of operations (discount, VAT, tip), it's often cleaner to
use the fractional form of the percentage instead of the decimal (e.g., 1/5
instead of 0.20 ).
This simplifies chain multiplication if you need to solve for the initial value
X in an expression like X multiplied by (4/5) multiplied by (11/10) equals Final
Total .
The Complement to 100% (or 1)
In mixture or filling problems, focus on what is missing or what is not present.
If 70% is water, 30% is something else.
The calculation error is often found in not using the remaining percentage when
performing successive operations (e.g., if 20% is removed, only 80% remains for
the next operation).
Difficult Level Percentage Exercises
Learn to calculate percentages step by step with difficult level examples and real-world cases.
Exercise 1 — Successive Discount for Target Price
An item was originally sold for S/400. The store wants to apply three successive discounts: 10%, 20%, and a third discount (X%) so that the final price is S/252. What must the percentage of the third discount (X%) be?
| Initial Price | S/400 |
|---|---|
| Final Price | S/252 |
| Known Discounts | 10% and 20% |
Price after 2 disc. = 400 × 0.90 × 0.80 = 288
Missing Discount Amount = 288 - 252 = 36
X% = (36 / 288) × 100 = 12.5%
Exercise 2 — Missing Score for a Target Average
A student needs to obtain a final grade of 85% in a course. The final exam is worth 60% of the total grade, and previous assignments are worth 40%. If they already obtained 80% on the previous assignments, what minimum score must they obtain on the final exam to reach the 85% goal?
| Total Goal | 85% |
|---|---|
| Final Exam Weight (X) | 60% |
| Previous Assignments Grade | 80% (Weight 40%) |
0.85 = (0.40 × 0.80) + (0.60 × X)
0.85 = 0.32 + 0.60X
0.53 = 0.60X
X = 0.53 / 0.60 ≈ 0.8833
Exercise 3 — Find the Initial Value with Known Variation
The price of a raw material increased by 40% in the last quarter. It is known that the current price (Final Value) is S/280 per unit. What was the original price (Initial Value) of the raw material before the increase?
| Final Value ($V_f$) | S/280 |
|---|---|
| Variation | +40% |
| Initial Value ($V_i$) | Unknown (X) |
1 + 0.40 = 1.40
Vf = Vi × Factor
280 = Vi × 1.40
Vi = 280 / 1.40 = 200
Exercise 4 — Undoing the Profit to Recoup the Cost
A seller set the price of a product by applying a 50% profit over its original cost. If they fail to sell it at that price, what percentage discount must they apply to the current selling price so that the final price is exactly equal to the original cost?
| Profit on Cost | 50% (Factor 1.50) |
|---|---|
| Original Cost ($C$) | Assume 100 (or 1) |
| Selling Price ($V$) | 150 (or 1.5) |
Amount to Reduce = Sale - Cost = 150 - 100 = 50
Discount % = (Amount to Reduce / Selling Price) × 100
= (50 / 150) × 100 ≈ 33.33%
Exercise 5 — Years to Double an Investment
An initial investment of S/10,000 grows at a fixed annual compound rate of 7%. Approximately how many years will it take for the investment to double (reach S/20,000)? (Note: This calculation requires logarithms, but can be estimated using the Rule of 72 or by iteration).
| Initial Capital ($C_i$) | S/10,000 |
|---|---|
| Final Capital ($C_f$) | S/20,000 |
| Growth Rate | 7% (Factor 1.07) |
Factor = Cf / Ci = 20000 / 10000 = 2
Years ≈ 72 / 7 ≈ 10.28 years
$2 = (1.07)^t$ $\rightarrow$ $t = \log_{1.07}(2) \approx 10.24$ years
Exercise 6 — Cumulative Error in Area Calculation
An engineer measures the side of a square plot of land and makes an error, measuring 3% less than the actual measure. If this incorrect measure is used to calculate the land's area, what will be the percentage of error in the area calculation?
| Error in the Side ($L$) | -3% (Factor 0.97) |
|---|---|
| Actual Area ($L^2$) | Assume $100^2 = 10,000$ (or $1^2 = 1$) |
Area Factor = (Side Factor)² = (0.97)² = 0.9409
Measured Area = 100% × 0.9409 = 94.09%
Error = 100% - 94.09% = 5.91%
Exercise 7 — Sample Size to Reach 50%
At an event, there are 300 men and 200 women. How many additional women must enter the event so that the percentage of women is 50% of the total attendees? (The number of men does not change).
| Men (Fixed) | 300 |
|---|---|
| Initial Women | 200 |
| Women Target | 50% of Total |
300 Men = 50% of Total
Final Total = (300 / 0.50) = 600 attendees
Final Women = 600 - 300 = 300
Additional Women = 300 - 200 = 100
Exercise 8 — Find the Annual Growth Rate
A company's annual turnover increased from S/500,000 to S/605,000 over a period of two years. If the growth was constant and compounded annually, what was the annual growth rate (r%)?
| Initial ($C_i$) | S/500,000 |
|---|---|
| Final ($C_f$) | S/605,000 |
| Time ($t$) | 2 years |
$C_f = C_i \times (1+r)^2$ $\rightarrow$ $(1+r)^2 = C_f / C_i$
$(1+r)^2 = 605000 / 500000 = 1.21$
$1+r = \sqrt{1.21} = 1.10$
$r = 1.10 - 1 = 0.10$
Exercise 9 — Radioactive Material Decay
A laboratory has a 500-gram sample of a radioactive isotope. If this material decays at a constant rate of 5% annually, approximately how many years will it take for the sample to be reduced to half of its initial mass (250 grams)?
| Initial Mass ($M_i$) | 500 grams |
|---|---|
| Final Mass ($M_f$) | 250 grams |
| Decay Rate | 5% annually (Factor 0.95) |
Factor = Mf / Mi = 250 / 500 = 0.5
$0.5 = (0.95)^t$ $\rightarrow$ $t = \log_{0.95}(0.5)$
$t \approx 13.51$ years
Exercise 10 — Cubic Error in Volume Measurement
The volume of a cubic box was calculated incorrectly, resulting in a measured volume that is 11.45% less than the actual volume. If the error was only in the measurement of the cube's side (one-dimensional error), what was the percentage of error in the measurement of that side?
| Error in Volume | -11.45% |
|---|---|
| Measured Volume (Factor) | $1 - 0.1145 = 0.8855$ |
$V_{\text{measured}} / V_{\text{actual}} = 0.8855$
Side Factor ($L_{\text{factor}}$) = $\sqrt[3]{0.8855} \approx 0.9599$
$1 - 0.9599 = 0.0401$
Exercise 11 — Annual Logistics Surcharge Rate
The transportation cost of a product increased from $4.50 per unit to $5.832 per unit over a period of three years. If the logistics surcharge was a constant annual percentage over the previous year's cost, what was the annual surcharge rate (r%)?
| Initial Cost ($C_i$) | $4.50 |
|---|---|
| Final Cost ($C_f$) | $5.832 |
| Time ($t$) | 3 years |
$C_f = C_i \times (1+r)^3$ $\rightarrow$ $(1+r)^3 = C_f / C_i$
$(1+r)^3 = 5.832 / 4.50 = 1.296$
$1+r = \sqrt[3]{1.296} = 1.09$
$r = 1.09 - 1 = 0.09$
Exercise 12 — Net Change of Two Transactions
An investor buys a stock for $X. The next day, the stock goes up by 20%. On the third day, the stock drops by 20% compared to the second day's value. What is the net percentage change in the stock's value from the initial price ($X$)?
| Day 1 (Initial) | $X$ (Assume 100) |
|---|---|
| Day 2 (Increase) | +20% (Factor 1.20) |
| Day 3 (Decrease) | -20% (Factor 0.80) |
Net Factor = 1.20 × 0.80 = 0.96
Final Value = 100 × 0.96 = 96
Change = (Final / Initial) - 1 = (96 / 100) - 1 = -0.04
Exercise 13 — Determine Cost to Earn a 25% Profit
A retailer sells a product for S/400. Knowing that, by selling it at that price, they obtain a profit of 25% on the original cost ($C$). What was the original cost of that product?
| Selling Price ($V$) | S/400 |
|---|---|
| Profit on Cost | 25% |
| Original Cost ($C$) | Unknown (X) |
Sale = Cost × (1 + Profit %)
$400 = C \times (1 + 0.25)$
$400 = C \times 1.25$
$C = 400 / 1.25 = 320$
Exercise 14 — Compound Growth with Variable Rates
An economic indicator started at 100 points ($P_i$). The first year it grew by 10%. The second year it dropped by 5% (based on the first year's value). The third year it grew by 15% (based on the second year's value). What is the value of the indicator at the end of the third year?
| Starting Point | 100 |
|---|---|
| Year 1 | +10% (Factor 1.10) |
| Year 2 | -5% (Factor 0.95) |
| Year 3 | +15% (Factor 1.15) |
Net Factor = 1.10 × 0.95 × 1.15
Net Factor = 1.045 × 1.15 = 1.20175
Final Value = 100 × 1.20175 = 120.175
Exercise 15 — Find the Initial Concentration of a Solution
There are 20 liters of a syrup with an unknown sugar concentration ($X\%$). By adding 5 liters of pure water (0% sugar), the final concentration is reduced to 16%. What was the initial concentration ($X\%$) of the syrup?
| Initial Volume ($V_i$) | 20 liters |
|---|---|
| Added Volume (Water) | 5 liters |
| Final Concentration ($C_f$) | 16% |
$V_f = 20 + 5 = 25$ liters
Sugar = $V_f \times C_f = 25 \times 0.16 = 4$ liters
$C_i = \text{Sugar} / V_i = 4 / 20 = 0.20$
Exercise 16 — Percentage Profit on Sale
A company determines its selling prices by applying a profit of 40% on the original cost ($C$). If a product has a cost of S/100, what percentage of profit does that benefit represent on the final selling price?
| Original Cost ($C$) | S/100 |
|---|---|
| Profit on Cost | 40% |
| Unknown | % Profit on Sale |
Sale = 100 × (1 + 0.40) = S/140
Gross Profit = 140 - 100 = S/40
% Profit on Sale = (Profit / Sale) × 100
= (40 / 140) × 100 ≈ 28.57%
Exercise 17 — Time Variation with Staff Change
A team of 10 programmers can complete a project in 20 days . If the company needs to complete the project 25% faster , and it is known that the productivity of each new programmer is 20% less than that of the original ones, how many additional programmers must be hired?
| Initial ($P_i$) | 10 programmers |
|---|---|
| Initial Time ($T_i$) | 20 days |
| Target Time ($T_f$) | -25% (15 days) |
Target Time: 20 × (1 - 0.25) = 15 days
Total Work (Initial): 10 prog. × 20 days = 200 units
New Productivity: 1 - 0.20 = 0.8 units/day (compared to original)
Total Needed = 200 units / 15 days ≈ 13.33 programmers
Exercise 18 — Original Cost for Neutral Sale
An investor sells two paintings for S/4,000 each , totaling S/8,000 in sales. On the first painting, they obtained a 25% profit over the cost. For the total net profit of both sales to be exactly 0 (no overall profit or loss), what must have been the original cost of the second painting?
| Sale Price of Each Painting | S/4,000 |
|---|---|
| Total Net Profit | S/0 |
| Profit Painting 1 (on Cost) | +25% |
$4000 = C_1 \times 1.25$ $\rightarrow$ $C_1 = 4000 / 1.25 = 3,200$
Profit 1 = $4000 - 3200 = 800$
Loss 2 = Profit 1 = 800$
Cost 2 = $4000 + 800 = 4,800$
Exercise 19 — Percentage of a Subgroup relative to the Rest
In a factory, there are 400 employees . 40% are technicians and 60% are operators . If 15% of the technicians are women , and 10% of the operators are women , what percentage of all male employees do the female operators represent?
| Total Employees | 400 |
|---|---|
| % Men (Total) | Unknown (New Base) |
| Female Operators | Unknown (Part) |
Technicians (40% of 400): 160; Female Technicians: $160 \times 0.15 =
24$
Operators (60% of 400): 240; Female Operators : $240 \times 0.10 =
24
Total Women: $24 + 24 = 48$
Total Men: $400 - 48 = 352
% = (Female Operators / Total Men) × 100
% = (24 / 352) × 100 ≈ 6.82%
Exercise 20 — Period with Constant Rate
An insect population grows at a constant rate of 20% each month . If the population went from 500 to 720 in $X$ months. What is the duration of that period ( $X$ months )?
| Initial ($P_i$) | 500 |
|---|---|
| Final ($P_f$) | 720 |
| Monthly Rate | +20% (Factor 1.20) |
$P_f = P_i \times (1.20)^X$ $\rightarrow$ $(1.20)^X = P_f / P_i$
$(1.20)^X = 720 / 500 = 1.44$
$1.20^1 = 1.20$; $1.20^2 = 1.44$
$X = 2$