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CMS Math Olympiad · 6th Grade

Percentages and Proportions

Discounts, increases, and parts of a whole. Percentages are the language of everyday math, and they appear in olympiads all the time.

What is a percentage?

Definition

\(1\% = \dfrac{1}{100}\)

“Percent” literally means “per hundred.” 25% = 25 out of 100 = one quarter.

Useful percentages to remember

\(50\% = \dfrac{1}{2}\)\(25\% = \dfrac{1}{4}\) \(20\% = \dfrac{1}{5}\)\(10\% = \dfrac{1}{10}\) \(75\% = \dfrac{3}{4}\)\(33\tfrac{1}{3}\% = \dfrac{1}{3}\)

Try it

\(40\%\) of 250 equals:

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Three types of problems

Almost every percentage problem is one of three types:

Type 1 — Find the part

What is 30% of 200?

\(200 \times 0.30 = 60\). Or: \(\dfrac{30}{100} \times 200 = 60\).

Type 2 — Find the whole

20% of the students is 50 people. How many students are there in total?

If \(20\% = 50\), then \(1\% = \dfrac{50}{20} = 2.5\). So \(100\% = 250\).

Type 3 — Find the percentage

Out of 80 students, 20 are top students. What percentage is that?

\(\dfrac{20}{80} = 0{,}25 = 25\%\).

CMS problem 2025 — Q2

20% of students do NOT use TokTik. The number who do use it is 200. How many do not use it?

200 = 80% (because 100% − 20% = 80%).
\(1\% = \dfrac{200}{80} = 2.5\). Therefore \(20\% = 50\).

Try it

15% of the price of a book is 12 UAH. The full price of the book is:

Try it

In a class of 40 students, 14 got top marks. The percentage of top students is:

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Increase and decrease

Main rule

Increase by \(p\%\): multiply by \(\left(1 + \dfrac{p}{100}\right)\)
Decrease by \(p\%\): multiply by \(\left(1 - \dfrac{p}{100}\right)\)

Example — discount

An item costs 800 UAH. The discount is 25%. What is the new price?

\(800 \times (1 - 0{,}25) = 800 \times 0{,}75 = 600\) UAH.

Or: discount = \(800 \times 0.25 = 200\). New price = \(800 - 200 = 600\).

Typical trap

Increase by 20%, then decrease by 20% — this does NOT return you to the starting value!

Start with 100. Increase by 20%: \(100 \times 1.2 = 120\).
Decrease by 20%: \(120 \times 0.8 = 96\). This is 4% less than the original!

Try it

A price increased by 50% and then decreased by 50%. Compared with the starting price, the result is:

Let the original value be 100.
+50%: \(100 \times 1{,}5 = 150\).
−50%: \(150 \times 0{,}5 = 75\).
\(75\) instead of \(100\) means 25% less.

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Change of area

This is a favourite CMS topic! If the side lengths of a figure change by percentages, the area changes differently.

Key fact

If each side increases by \(p\%\), the area does not increase by \(p\%\), but by more!

New area = \((1 + \dfrac{p}{100})^2 \times S\)

Test problem — Q23

The sides of a rectangle were increased by 10%. By what percentage did the area increase?

Multiplier for each side: \(1.1\).

New area: \(1.1 \times 1.1 = 1.21\) of the original.

Increase: \(1.21 - 1 = 0.21 = \mathbf{21\%}\).

Quick calculation formula

If both side lengths increase by \(p\%\):
The area increases by about \(2p + \dfrac{p^2}{100}\) percent.

For \(p = 10\%\): \(2 \times 10 + \dfrac{100}{100} = 20 + 1 = 21\%\). Exactly!

Try it

The sides of a square were increased by 20%. The area increased by:

\(1.2 \times 1.2 = 1.44\). Increase: \(44\%\).
Or: \(2 \times 20 + \dfrac{400}{100} = 40 + 4 = 44\%\).

Try it

One side of a rectangle was increased by 50%, the other decreased by 50%. The area:

\(1.5 \times 0.5 = 0.75\). The area becomes 75% of the original → it decreased by 25%.

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Proportions and ratios

A proportion is an equality of two fractions: \(\dfrac{a}{b} = \dfrac{c}{d}\).

A ratio \(a : b\) means that for every \(a\) parts of one quantity there are \(b\) parts of the other.

Example — divide in a ratio

180 sweets are divided between Alina and Bohdan in the ratio 2 : 3. How many did Bohdan get?

Total parts: \(2 + 3 = 5\).

One part: \(180 \div 5 = 36\).

Bohdan (3 parts): \(36 \times 3 = 108\).

CMS problem 2024 — Q5

A book has Arithmetic and Geometry sections. Geometry is \(\dfrac{1}{3}\) of Arithmetic. What fraction of the book is Arithmetic?

Let Arithmetic = 3 parts, Geometry = 1 part.
Total = 4 parts. Arithmetic = \(\dfrac{3}{4}\) of the book.

Try it

240 UAH is divided in the ratio 3 : 5. The larger part is:

Try it

In a class the ratio of boys to girls is 3 : 2. There are 35 students in total. How many girls are there?

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Olympiad practice

Problem 1 · Easy

In a class, 25% of students wear glasses. If there are 32 students, how many wear glasses?

Problem 2 · Easy

An item cost 400 UAH. The price was reduced by 15%. The new price is:

Problem 3 · Medium

After a 25% price increase, an item costs 500 UAH. The original price is:

500 = 125% of the original.
\(1\% = \dfrac{500}{125} = 4\). Therefore \(100\% = 400\) UAH.

Problem 4 · Medium

The side of a square was increased by 30%. By what percentage did the area increase?

\(1.3 \times 1.3 = 1.69\). The area increased by \(69\%\).
Or: \(2 \times 30 + \dfrac{900}{100} = 60 + 9 = 69\%\).

Problem 5 · Medium

A mixture contains water and juice in the ratio 7 : 3. How much juice is in 500 ml of the mixture?

Total parts: \(7 + 3 = 10\).
One part: \(500 \div 10 = 50\) ml.
Juice (3 parts): \(50 \times 3 = 150\) ml.

Problem 6 · Medium+

A price was increased by 10% twice. The total increase is:

\(1.1 \times 1.1 = 1.21\). Total increase = 21%.
Two times 10% is not 20%! The second 10% is calculated from the larger number.

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Cheat sheet — Percentages and Proportions

Find the part: number \(\times \dfrac{p}{100}\)
Find the whole: part \(\div \dfrac{p}{100}\)
Find the percentage: \(\dfrac{\text{part}}{\text{whole}} \times 100\%\)
Increase by \(p\%\): \(\times (1 + \dfrac{p}{100})\)
Decrease by \(p\%\): \(\times (1 - \dfrac{p}{100})\)
Area when side lengths change by \(p\%\): the area changes by \(\approx 2p + \dfrac{p^2}{100}\)%
A proportion \(a : b\): one part = whole \(\div (a+b)\)
Trap: +20% then −20% is not the original value!

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Percentages and Proportions

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