Converting a score like 11 out of 15 to a percentage is one of the most practical mathematics skills a student can have — and it is also directly tested on the Gauss math contest in problems involving percentages, fractions, and decimals. This guide explains exactly how to find the percentage of 11 out of 15, works through other commonly searched score conversions including 36 out of 40, 25 out of 30, 35 out of 40, 43 out of 50, and 18 out of 25, and then builds from these foundations to the more complex percentage problems that appear in Gauss past papers. Whether your child needs to convert a test score, solve a percentage word problem, or answer a Gauss contest question involving percentages, decimals and fractions, this guide covers all of it with full worked examples.
Signing up for the Gauss contest or thinking about registering? Check out: Gauss Contest: What Canadian Students Need to Know Before They Compete.
What is a percentage and how does it work?
A percentage is a way of expressing a number as a fraction of 100. The word percent comes from the Latin per centum meaning out of one hundred. When you say 75%, you mean 75 out of every 100 — or equivalently, the fraction 75/100.
The core idea behind every percentage calculation is the same: you are scaling a fraction so the denominator becomes 100.
The percentage formula:
Percentage = (part / whole) x 100
This formula works for every score conversion, every percentage word problem, and every percentage question on the Gauss math contest. Memorise it and apply it consistently.
The relationship between percentages, decimals and fractions
Percentages, decimals and fractions are three ways of expressing the same value. Converting fluently between all three is one of the core skills tested on the Gauss contest.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 1/10 | 0.1 | 10% |
| 3/10 | 0.3 | 30% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333… | 33.3…% |
| 2/3 | 0.666… | 66.6…% |
Converting fraction to percentage: multiply by 100. Converting percentage to fraction: divide by 100 and simplify. Converting decimal to percentage: multiply by 100. Converting percentage to decimal: divide by 100.
For a full set of practice problems converting between all three forms, our guide to equivalent fractions covers the proportional reasoning that underpins these conversions in depth — see Equivalent Fractions Worksheet: Practice Problems and Examples.
Percentage of 11 out of 15
Finding the percentage of 11 out of 15 is a direct application of the percentage formula.
Method:
Percentage = (part / whole) x 100 = (11 / 15) x 100 = 0.7333… x 100 = 73.33…%
Rounded to the nearest whole number: 73% Rounded to one decimal place: 73.3%
As a fraction: 11/15 cannot be simplified further since 11 is prime and does not divide 15.
As a decimal: 11 ÷ 15 = 0.7333… (the 3 repeats indefinitely)
Why 11 out of 15 does not give a clean percentage
Unlike fractions whose denominators are factors of 100 (such as 4, 5, 10, 20, 25, 50), the fraction 11/15 does not convert to a terminating decimal. This is because 15 = 3 x 5, and 3 is not a factor of 100. Whenever the denominator of a simplified fraction contains a prime factor other than 2 or 5, the decimal expansion will repeat rather than terminate.
For the Gauss math contest, you need to be comfortable working with both terminating and repeating decimals. Recognising that 1/3 = 0.333…, 2/3 = 0.666…, 1/6 = 0.1666…, and 1/15 = 0.0666… will help you work with fractions that have 3 in the denominator more efficiently.
Quick percentage reference — what is 11 out of 15?
| Score | Fraction | Decimal | Percentage |
|---|---|---|---|
| 11 out of 15 | 11/15 | 0.7333… | 73.3% |
| 12 out of 15 | 4/5 | 0.8 | 80% |
| 13 out of 15 | 13/15 | 0.8666… | 86.7% |
| 14 out of 15 | 14/15 | 0.9333… | 93.3% |
| 15 out of 15 | 1 | 1.0 | 100% |
36 out of 40 as a percentage
36 out of 40 is a cleaner calculation than 11 out of 15 because 40 is a factor-of-100-friendly number.
Method 1 — formula: Percentage = (36 / 40) x 100 = 0.9 x 100 = 90%
Method 2 — scaling the fraction: 36/40 = 9/10 (divide both by 4). 9/10 = 90/100 = 90%
Method 3 — recognising the fraction: 36/40 simplifies to 9/10, which is a benchmark fraction. 9/10 = 90% is worth knowing directly.
Answer: 36 out of 40 = 90%
H3: 35 out of 40 as a percentage
Method: 35/40 = 7/8 (divide both by 5). 7/8 as a percentage: (7/8) x 100 = 700/8 = 87.5%
Answer: 35 out of 40 = 87.5%
Alternatively: 35/40 = 0.875. Multiply by 100 to get 87.5%.
Score comparison table out of 40
| Score | Simplified fraction | Percentage |
|---|---|---|
| 32/40 | 4/5 | 80% |
| 33/40 | 33/40 | 82.5% |
| 34/40 | 17/20 | 85% |
| 35/40 | 7/8 | 87.5% |
| 36/40 | 9/10 | 90% |
| 37/40 | 37/40 | 92.5% |
| 38/40 | 19/20 | 95% |
| 39/40 | 39/40 | 97.5% |
| 40/40 | 1 | 100% |
43 out of 50 as a percentage
50 is one of the most convenient denominators for percentage conversion because 50 x 2 = 100.
Method: 43/50 x 2/2 = 86/100 = 86%
Multiplying both numerator and denominator by 2 scales the fraction directly to hundredths — no division needed.
Answer: 43 out of 50 = 86%
Why 50 is easy to convert
Any fraction with denominator 50 converts to a percentage instantly by doubling the numerator. This is because 100/50 = 2, so multiplying numerator and denominator by 2 always gives a denominator of 100.
- 27/50 = 54%
- 33/50 = 66%
- 41/50 = 82%
- 47/50 = 94%
Similarly, fractions with denominator 25 multiply by 4 to reach 100, fractions with denominator 20 multiply by 5, and fractions with denominator 10 multiply by 10.
25 out of 30 as a percentage
25 out of 30 involves the factor 3 in the denominator, so it will produce a repeating decimal.
Method: 25/30 = 5/6 (divide both by 5). (5/6) x 100 = 500/6 = 83.333…%
Rounded to one decimal place: 83.3% Rounded to the nearest whole number: 83%
Answer: 25 out of 30 = 83.3%
Score comparison table out of 30
| Score | Simplified fraction | Percentage |
|---|---|---|
| 24/30 | 4/5 | 80% |
| 25/30 | 5/6 | 83.3% |
| 26/30 | 13/15 | 86.7% |
| 27/30 | 9/10 | 90% |
| 28/30 | 14/15 | 93.3% |
| 29/30 | 29/30 | 96.7% |
| 30/30 | 1 | 100% |
18 out of 25 as a percentage
25 is one of the easiest denominators to convert because 25 x 4 = 100.
Method: 18/25 x 4/4 = 72/100 = 72%
Answer: 18 out of 25 = 72%
Score comparison table out of 25
| Score | Multiply by | Percentage |
|---|---|---|
| 15/25 | x4 = 60/100 | 60% |
| 17/25 | x4 = 68/100 | 68% |
| 18/25 | x4 = 72/100 | 72% |
| 20/25 | x4 = 80/100 | 80% |
| 22/25 | x4 = 88/100 | 88% |
| 24/25 | x4 = 96/100 | 96% |
Percentages, decimals and fractions worksheet
These problems cover the full range of percentage, decimal, and fraction conversions that appear on the Gauss math contest. They are organised by increasing difficulty.
Level 1 — direct conversions
Problem 1: Convert 3/5 to a percentage.
Solution: 3/5 x 100 = 60%
Answer: 60%
Problem 2: Convert 45% to a fraction in simplest form and to a decimal.
Solution: Fraction: 45/100 = 9/20 (divide by 5). Decimal: 45 ÷ 100 = 0.45.
Answer: 9/20 and 0.45
Problem 3: What percentage is 18 out of 25?
Solution: 18/25 x 4/4 = 72/100 = 72%.
Answer: 72%
Problem 4: Express 0.375 as a fraction and as a percentage.
Solution: 0.375 = 375/1000 = 3/8 (divide by 125). Percentage: 0.375 x 100 = 37.5%.
Answer: 3/8 and 37.5%
Problem 5: Which is larger — 7/12 or 58%?
Solution: Convert 7/12 to percentage: (7/12) x 100 = 700/12 = 58.333…% 58.333…% > 58%.
Answer: 7/12 is larger
Level 2 — percentage calculations
Problem 6: What is 35% of 80?
Solution: 35% of 80 = 0.35 x 80 = 28.
Answer: 28
Problem 7: A student scores 36 out of 40 on a test. What percentage did they score?
Solution: 36/40 x 100 = 90%.
Answer: 90%
Problem 8: A price increases from $40 to $52. What is the percentage increase?
Solution: Increase = 52 – 40 = $12. Percentage increase = (12/40) x 100 = 30%.
Answer: 30%
Problem 9: A class has 30 students. 18 are girls. What percentage are boys?
Solution: Girls: 18/30 = 60%. Boys: 100% – 60% = 40%. Or: boys = 12, 12/30 x 100 = 40%.
Answer: 40%
Problem 10: A jacket costs $120 after a 25% discount. What was the original price?
Solution: After discount, the price represents 75% of original (100% – 25% = 75%). 75% of original = $120. Original = 120 / 0.75 = $160.
Answer: $160
Key insight: Working backwards from a discounted price is a common Gauss problem type. If a price after a percentage reduction is given, divide by (1 – reduction rate) to find the original. This is a reverse percentage calculation.
Level 3 — Gauss style percentage problems
Problem 11 — Gauss style
In a class of 40 students, 60% play sport and 45% play a musical instrument. If 20% play both, what percentage play neither?
Solution: Play sport: 60% Play instrument: 45% Play both: 20%
Play at least one: 60% + 45% – 20% = 85% (subtract the overlap to avoid double-counting).
Play neither: 100% – 85% = 15%.
Answer: 15%
Key insight: The inclusion-exclusion principle — adding two groups and subtracting the overlap — appears regularly on the Gauss contest in percentage, fraction, and counting problems. For more on counting techniques that use the same principle, see AMC 8 Counting and Probability: Key Rules and Real Problems.
Problem 12 — Gauss style
A store sells apples at $3 for 5. It increases the price by 20% and a customer buys 15 apples. How much do they pay?
Solution: Original price per apple: $3 ÷ 5 = $0.60. New price after 20% increase: $0.60 x 1.20 = $0.72 per apple. Cost for 15 apples: 15 x $0.72 = $10.80.
Answer: $10.80
Key insight: Percentage increase problems multiply by (1 + rate) rather than adding the percentage amount separately. Multiplying by 1.20 gives the new price in one step rather than calculating 20% of $0.60 and adding it. This is faster and less prone to error.
Problem 13 — Gauss style
A rectangle’s length is increased by 10% and its width is decreased by 10%. What is the percentage change in area?
Solution: Let original length = L and width = W. Original area = L x W.
New length = 1.1L. New width = 0.9W. New area = 1.1L x 0.9W = 0.99LW.
Change = 0.99LW – LW = -0.01LW. Percentage change = -0.01/1 x 100 = -1%.
Answer: The area decreases by 1%
Key insight: This is a classic Gauss-style trap. Students who assume that a 10% increase and 10% decrease cancel out get 0% — the wrong answer. The actual change is -1% because 1.1 x 0.9 = 0.99, not 1.0. Percentage changes do not simply cancel.
Problem 14 — Gauss style
In a school, 40% of students are in Grade 7 and 60% are in Grade 8. Of the Grade 7 students, 25% take an art class. Of the Grade 8 students, 30% take an art class. What percentage of all students take an art class?
Solution: Grade 7 students taking art: 40% x 25% = 10% of all students. Grade 8 students taking art: 60% x 30% = 18% of all students.
Total taking art: 10% + 18% = 28%.
Answer: 28%
Key insight: Weighted percentage problems require multiplying each group’s percentage by the size of that group as a proportion of the whole. This is the weighted average concept — a very common Gauss contest technique.
Problem 15 — Gauss style
A school’s student population increased by 25% in one year and then decreased by 20% the following year. What was the overall percentage change over the two years?
Solution: Let original population = 100 (using 100 as a base makes percentage calculations easier).
After 25% increase: 100 x 1.25 = 125. After 20% decrease: 125 x 0.80 = 100.
Overall change: 100 – 100 = 0.
Answer: 0% — the population returned to its original value
Key insight: A 25% increase followed by a 20% decrease returns to the original value because 1.25 x 0.80 = 1.00 exactly. This is another classic Gauss trap — the percentages are different going up and down because they apply to different base values. Students who add and subtract percentages directly get 25% – 20% = 5% increase, which is wrong.
Problem 16 — hardest
A survey of 200 families found that 35% own a dog, 25% own a cat, and 15% own both. What is the probability that a randomly selected family owns a dog or a cat but not both?
Solution: Own dog only: 35% – 15% = 20%. Own cat only: 25% – 15% = 10%. Own dog or cat but not both: 20% + 10% = 30%.
Number of families: 30% of 200 = 60.
Answer: 30% (60 families)
Key insight: “But not both” problems require subtracting the overlap from each individual group before adding. This is a precise application of the inclusion-exclusion principle that appears on Gauss Part B and C questions.
Converting fractions to percentages — quick methods by denominator
Rather than always using the formula, recognising which scaling method works for each denominator saves significant time under Gauss contest conditions.
| Denominator | Scaling method | Example |
|---|---|---|
| 2 | Multiply by 50 | 1/2 = 50% |
| 4 | Multiply by 25 | 3/4 = 75% |
| 5 | Multiply by 20 | 4/5 = 80% |
| 8 | Multiply by 12.5 | 5/8 = 62.5% |
| 10 | Multiply by 10 | 7/10 = 70% |
| 20 | Multiply by 5 | 13/20 = 65% |
| 25 | Multiply by 4 | 18/25 = 72% |
| 40 | Multiply by 2.5 | 36/40 = 90% |
| 50 | Multiply by 2 | 43/50 = 86% |
| 3 | Divide and multiply — gives repeating decimal | 2/3 = 66.7% |
| 6 | Divide and multiply | 5/6 = 83.3% |
| 15 | Divide and multiply | 11/15 = 73.3% |
Denominators that are factors of 100 (2, 4, 5, 10, 20, 25, 50) always give terminating decimals and exact percentages. Denominators containing the prime factor 3 (3, 6, 9, 12, 15) always give repeating decimals.
Percentage benchmarks worth memorising
Students who have these benchmark fractions and their percentage equivalents memorised can solve many Gauss contest problems significantly faster than students who calculate from scratch.
| Fraction | Percentage | Fraction | Percentage |
|---|---|---|---|
| 1/2 | 50% | 1/6 | 16.7% |
| 1/3 | 33.3% | 5/6 | 83.3% |
| 2/3 | 66.7% | 1/8 | 12.5% |
| 1/4 | 25% | 3/8 | 37.5% |
| 3/4 | 75% | 5/8 | 62.5% |
| 1/5 | 20% | 7/8 | 87.5% |
| 2/5 | 40% | 1/10 | 10% |
| 3/5 | 60% | 3/10 | 30% |
| 4/5 | 80% | 7/10 | 70% |
How percentages appear on the Gauss math contest
Percentages on the Gauss appear in five main formats. Knowing which format a question uses before starting to solve it prevents applying the wrong method.
Direct conversion: Convert a fraction or score to a percentage, or vice versa. These are the most straightforward and appear in Part A.
Finding a percentage of a quantity: “What is 35% of 120?” Multiply the quantity by the decimal form of the percentage. These appear across Parts A and B.
Percentage increase and decrease: “A price increases by 15%. What is the new price?” Multiply by (1 + rate) for increases, (1 – rate) for decreases. Common in Parts A and B. For more on rate and proportion problems that combine with percentages on the Gauss, see our unit rate guide.
Reverse percentage: “After a 20% increase the price is $84. What was the original price?” Divide by (1 + rate). Common in Parts B and C.
Multi-step percentage problems: Combining percentage increases or decreases in sequence, weighted averages, or inclusion-exclusion. These appear in Parts B and C and are where most marks are lost.
For percentage problems involving algebraic setup — where the percentage relationship must be expressed as an equation and solved — the equation-solving techniques covered in our grade 8 algebra guide are directly relevant. See Grade 8 Algebra Worksheets: Practice Problems for the Gauss Contest] for worked examples.
Frequently Asked Questions
What is the percentage of 11 out of 15? 11 out of 15 as a percentage is 73.3% (to one decimal place) or approximately 73%. To calculate it, divide 11 by 15 to get 0.7333… and multiply by 100. The result is a repeating decimal because 15 contains the prime factor 3, which is not a factor of 100.
What is 36 out of 40 as a percentage? 36 out of 40 is 90%. Simplify 36/40 to 9/10 by dividing both by 4, then multiply by 100 to get 90%. Alternatively, divide 36 by 40 to get 0.9, then multiply by 100.
What is 35 out of 40 as a percentage? 35 out of 40 is 87.5%. Simplify 35/40 to 7/8, then multiply by 100: (7/8) x 100 = 87.5%. Alternatively, 35 ÷ 40 = 0.875, and 0.875 x 100 = 87.5%.
What is 43 out of 50 as a percentage? 43 out of 50 is 86%. Because 50 x 2 = 100, simply double the numerator: 43 x 2 = 86. So 43/50 = 86/100 = 86%.
What is 25 out of 30 as a percentage? 25 out of 30 is approximately 83.3%. Simplify 25/30 to 5/6, then multiply by 100: (5/6) x 100 = 83.333…%. Because 30 contains the prime factor 3, the decimal repeats.
What is 18 out of 25 as a percentage? 18 out of 25 is 72%. Because 25 x 4 = 100, multiply both numerator and denominator by 4: 18 x 4 = 72. So 18/25 = 72/100 = 72%.
How do percentages, decimals and fractions relate to each other? They are three different ways of expressing the same value. To convert a fraction to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100. To convert a decimal to a fraction, write it over the appropriate power of 10 and simplify. Building fluency converting between all three forms is essential for the Gauss math contest.
How do percentages appear on the Gauss math contest? Percentage questions on the Gauss appear as direct conversions, percentage of a quantity calculations, percentage increase and decrease problems, reverse percentage problems (finding the original value before a change), and multi-step problems combining percentages with fractions or algebra. Part C questions often involve percentage traps — for example, a 25% increase followed by a 20% decrease does not return to the original value.
