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Ratio To Percentage

Here we will learn about ratio to percentage, including how to convert from ratios to percentages and how to solve word problems involving converting ratios to percentages.

There are also ratio to percentage worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is ratio to percentage?

Ratio to percentage is when you use a given ratio to calculate a percentage.

A ratio tells us how much there is of one thing in relation to another thing.

For example, if Olivia and Dean share some sweets in the ratio 3:2 then for every 3 sweets Olivia gets, Dean gets 2. You can use this information to write the percentage of sweets Olivia and Dean each get. 

To find the percentage of sweets they each get, first convert the ratio into fractions. 

Step-by-step guide: Ratio to fractions

The ratio 3:2 has 5 parts, so the fractions are \frac{3}{5} : \frac{2}{5}.

Olivia gets three fifths of the sweets and Dean gets two fifths. You can convert these fractions to percentages.

You may be able to recognise what the fractions are as percentages or you may need to use long division to help convert your fractions.

Step-by-step guide: Fractions to percentages

\frac{3}{5}=60\%, so Olivia gets 60\% of the sweets.

\frac{2}{5}=40\%, so Dean gets 40\% of the sweets.

How to convert a ratio to a percentage

In order to convert a ratio to a percentage:

Add the parts of the ratio for the denominator of the fractions.

Convert each part of the ratio to a fraction.

Convert the fractions to percentages.

Explain how to convert a ratio to a percentage

Ratio to percentage worksheet

Get your free ratio to percentage worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on ratio

Ratio to percentage is part of our series of lessons to support revision on ratio . You may find it helpful to start with the main ratio lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• How to work out ratio  
• Simplifying ratios
• Ratio to fraction
• Dividing ratios
• Ratio problem solving
• Ratio scale

Ratio to percentage examples

Example 1: converting a two part ratio to percentages.

The ratio of blue counters to red counters is 3:1. Write the ratio as percentages.

3+1=4. There are 4 parts in total. The denominator is 4.

2 Convert each part of the ratio to a fraction.

3:1 becomes \frac{3}{4}:\frac{1}{4} .

3 Convert the fractions to percentages.

\frac{3}{4} and \frac{1}{4} are common fractions for which you need to know the percentage conversions.

Example 2: converting a two part ratio to percentages

The ratio of green counters to yellow counters is 3:5. What percentage of the counters are green?

3+5=8. There are 8 parts in total. The denominator is 8.

3:5 becomes \frac{3}{8}:\frac{5}{8}.

0.375=37.5\% and 0.625=62.5\% therefore,

\frac{3}{8}:\frac{5}{8}=37.5\%:62.5\% .

Order is important in ratios and in this ratio the number of green counters is first.

Therefore the percentage of counters that are green is 37.5\%.

Example 3: converting a three part ratio to percentages

A garden centre sells three types of bulbs; daffodil, tulip and lily, in the ratio 6:9:5. What percentage of the bulbs sold are tulips?

6 + 9 + 5 = 20. There are 20 parts in total. The denominator is 20.

6:9:5 becomes \frac{6}{20}:\frac{9}{20}:\frac{5}{20}.

Writing each fraction with a denominator of 100,

\frac{6}{20}:\frac{9}{20}:\frac{5}{20}=\frac{30}{100}:\frac{45}{100}:\frac{25}{100} .

\frac{6}{20}:\frac{9}{20}:\frac{5}{20}=30\%:45\%:25\%

In this question we are interested in the number of tulips sold, which is the middle value. The percentage that are tulips is 45\%.

Example 4: solving a problem involving ratio to percentages

In April 2022, Ben and Jacob shared some money in the ratio of their ages. Ben was born in June 2014 and Jacob in January 2019. What percentage of the money does Ben receive?

In April 2022 Ben is 7 and Jacob is 3. The ratio of their ages is 7:3.

7+3=10. There are 10 parts in total. The denominator is 10.

7:3 becomes \frac{7}{10}:\frac{3}{10}.

Ben receives 70\% of the money.

Example 5: solving a problem involving ratio to percentages

The ratio of adults to children in a park is 6:4.

One third of the adults are men. What percentage of the people in the park are women?

6 + 4 = 10. There are 10 parts in total. The denominator is 10.

6:4 becomes \frac{6}{10}:\frac{4}{10}.

You have worked out that 60\% of the people are adults. 

One third of the adults are men.

\frac{1}{3} of 60\% = 20\%

20\% of the people in the park are men and therefore 60-20=40\% of the people are women.

Common misconceptions

• Ratios and fractions confusion

For example, the ratio 2:3 is expressed as the fraction \frac{2}{3} and not \frac{2}{5}. This is a misunderstanding of the sum of the parts of the ratio. Be careful with what the question is asking as the denominator may be the part, or the whole amount.

• Ratio written in the wrong order

Order is important in ratio questions and you must maintain the original order throughout your working and answer.

Practice ratio to percentage questions

1. Convert the ratio 1:4 to percentages.

This ratio has 5 parts in total therefore

Converting to percentages,

2. The ratio of people who own a dog to people who don’t own a dog is 9:11. What percentage of people own a dog?

This ratio has 20 parts therefore

45\% of people own a dog.

3. The ratio of teachers to students on a school trip is 1:7. What percentage of those on the trip are students?

The total number of parts in this ratio is 8 therefore

87.5\% of those on the trip are students.

4. Sam, Katrina and Alex share some sweets in the ratio 11:8:6. What percentage of the sweets does Sam get?

The total number of parts in this ratio is 25 therefore

Sam gets 44\% of the sweets.

5. The ratio of the number of boys in a class to the number of girls in a class is 7:3. One fifth of the boys wear glasses. What percentage of the class are boys that wear glasses?

The total number of parts in this ratio is 10 therefore

70\% of the class are boys and one fifth of the boys wear glasses.

\frac{1}{5} of 70\% = 14\% .

14\% of the class are boys who wear glasses.

6. Tony and Anne share some money in the ratio 23:27.

Anne gives a quarter of her share to her friend James.

What percentage of the money does James get?

The total number of parts in this ratio is 50 therefore

Anne gets 46\% of the money and gives one quarter of her share to James.

\frac{1}{4} of 46\% = 11.5\% .

James gets 11.5\% of the money.

Ratio to percentage GCSE questions

1. Write the following ratio as percentages.

Circle your answer.

2. William and Matthew share some money in the ratio 3:7. Matthew gives two fifths of his money to Ellie.

Who receives the most money? Show how you decide.

William 30\%, Matthew 70\%

\frac{2}{5} of 70\%=28\% .

Matthew 42\%, Ellie 28\%

Matthew receives the most money.

3. The ratio of men to women working for an accountancy firm is 5:3.

One fifth of the women are managers and 10\% of the men are managers.

What percentage of all employees are managers?

For converting 5:3 to fractions or percentages, for example 62.5\%:37.5\%.

10\% of 62.5\%=6.25\%

\frac{1}{5} of 37.5\%=7.5\%

Learning checklist

You have now learned how to:

• Convert ratios to percentages and solve problems involving ratios and percentages

The next lessons are

• Compound measures
• Best buy maths

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Solving problems with percentages

• Price difference I
• Price difference II
• How many students?

To solve problems with percent we use the percent proportion shown in "Proportions and percent".

$$\frac{a}{b}=\frac{x}{100}$$

$$\frac{a}{{\color{red} {b}}}\cdot {\color{red} {b}}=\frac{x}{100}\cdot b$$

$$a=\frac{x}{100}\cdot b$$

x/100 is called the rate.

$$a=r\cdot b\Rightarrow Percent=Rate\cdot Base$$

Where the base is the original value and the percentage is the new value.

47% of the students in a class of 34 students has glasses or contacts. How many students in the class have either glasses or contacts?

$$a=r\cdot b$$

$$47\%=0.47a$$

$$=0.47\cdot 34$$

$$a=15.98\approx 16$$

16 of the students wear either glasses or contacts.

We often get reports about how much something has increased or decreased as a percent of change. The percent of change tells us how much something has changed in comparison to the original number. There are two different methods that we can use to find the percent of change.

The Mathplanet school has increased its student body from 150 students to 240 from last year. How big is the increase in percent?

We begin by subtracting the smaller number (the old value) from the greater number (the new value) to find the amount of change.

$$240-150=90$$

Then we find out how many percent this change corresponds to when compared to the original number of students

$$90=r\cdot 150$$

$$\frac{90}{150}=r$$

$$0.6=r= 60\%$$

We begin by finding the ratio between the old value (the original value) and the new value

$$percent\:of\:change=\frac{new\:value}{old\:value}=\frac{240}{150}=1.6$$

As you might remember 100% = 1. Since we have a percent of change that is bigger than 1 we know that we have an increase. To find out how big of an increase we've got we subtract 1 from 1.6.

$$1.6-1=0.6$$

$$0.6=60\%$$

As you can see both methods gave us the same answer which is that the student body has increased by 60%

Video lessons

A skirt cost $35 regulary in a shop. At a sale the price of the skirtreduces with 30%. How much will the skirt cost after the discount?

Solve "54 is 25% of what number?"

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Ratio and Percent Word Problems

Ratio word problems, using table with multiplication.

Using Double Number Lines

Using Tape Diagram

Using Graphs

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Unit 3: Ratios and percentages

Percent word problems.

• Solving percent problems (Opens a modal)
• Percent word problem: magic club (Opens a modal)
• Percent word problems: tax and discount (Opens a modal)
• Percent word problem: guavas (Opens a modal)
• Equivalent expressions with percent problems Get 3 of 4 questions to level up!
• Percent problems Get 3 of 4 questions to level up!
• Tax and tip word problems Get 3 of 4 questions to level up!
• Discount, markup, and commission word problems Get 3 of 4 questions to level up!

Unit conversion

• Ratios and measurement (Opens a modal)
• Ratios and units of measurement Get 3 of 4 questions to level up!

Rates with fractions

• Rates with fractions (Opens a modal)
• Rates with fractions Get 3 of 4 questions to level up!

Math Made Easy

Percentages, Ratios, and Proportions

This section covers:

Percentages and Percent Changes

Ratios and proportions, unit multipliers, using percentages with ratios.

• More Practice

Note :  For more problems with percents and ratios, see the Algebra Word Problems section .

Percentages are something you are probably quite familiar with because of your shopping habits, right? How many times have you been to the store when everything is 20% off? Do you notice how many people around you (adults, usually!) have no idea how to figure out what the sale price is? The easiest example of percentages is 50% off, which means that the item is half price.

Percentages really aren’t that difficult if you truly understand what they are. The word “percentage” comes from the word “per cent”, which means “per hundred” in Latin.  Remember that “per” usually means “over”. So “per cent” literally means “over 100 ” or “divided by 100 “. And remember what “of” typically means? I’ll write it again, since it’s so important:

OF  =  TIMES

When we say “ 20% off of something”, let’s translate it to “ 20 over (or divided by) 100 — then times the original price”, and that will be the amount we subtract from the original price.

Remember that we cannot use a percentage in math;  we need to turn it into a decimal.   To turn a percentage into a decimal, we move the decimal 2 places to the left (because we need to divide by 100 ), and if we need to turn a decimal back into a percentage, we move the decimal 2 places to the  right (because we need to multiply by 100 ).

I like to think of it this way : When we’re taking away the %, we are afraid of it, so we move 2 decimal places away from it (or to the left ). When we need to turn a number into a %, we like it, so we move 2 decimals towards it (or to the right ).

Let’s get back to our percentage example.  If there’s a dress we like for say $50 , and it’s 20% off (“off” means take-away or minus!), we’ll do the math to figure out the sales price.  This is called a percent change   problem.

Amount of sale : $ \displaystyle 20\%\,\,\text{of }\,\$50=.2\times \$50=\$10.\,\,\,\,\$50-\$10=\$40$. The dress would be $40 .

(See how we had to turn the 20% into a decimal by taking away the % sign and moving 2 decimals to the left, or away from it, since we didn’t like it?)

We could have also multiplied the original price by $ \displaystyle 80\%\,(100\%-20\%)$, or $ \displaystyle \frac{{80}}{{100}}$, since that’s what we’ll be paying if we get 20% off ( 100% full price minus 20% discount equals 80% discounted price):

Price of discounted dress : $ \displaystyle 80\%\,\,\text{of }\,\$50=.8\times \$50=\$40$. This method has fewer steps.

This shopping example is a percent decrease problem; the following is the formula for that. Make sure you relate this formula back to the example above.

$ \displaystyle \text{Newer}\,\,\text{lower}\,\,\text{price =}\,\,\text{original}\,\,\text{price}\,\,-\,\,\left( {\text{original}\,\,\text{price}\,\,\times \,\,\left. {\frac{{\text{percentage}\,\,\text{off}}}{{100}}} \right)} \right.$

$ \displaystyle \$50-\left( {\$50\,\,\times \,\,\left. {\frac{{20}}{{100}}} \right)} \right.\,\,=\,\,\$50-\$10=\$40$

Notice that we worked the math in the parentheses first (we will get to this in more detail later) .

Now let’s talk about a percent increase problem, which is also a percent change problem. A great example of a percent increase is the tax you pay on this dress. Tax is a percentage (usually) that you add on to what you pay so we can continue driving on the streets free and going to public school free.

If we need to add on 8.25% sales tax to the $40 that we are going to spend on the dress, we’ll have to know the percent increase formula, but let’s first figure it out without the formula.  Tax is the amount we have to add that is based on a percentage of the price that we’re paying for the dress.

The tax would be 8.25% or .0825 (remember – we don’t like the %, so we take it away and move away from it?) times the price of the dress and then add it back to the price of the dress.

Total price with tax:   $ \displaystyle \$50+(8.25\%\times 50)=\$50+(.0825\times 50)=\$50+\$4.125=\$54.125=\$54.13$.

Note that we rounded up to two decimal places, since we’re dealing with money. Note also that we did the math inside the parentheses first.

The total price of the dress would be $54.13 .

Here’s the formula:

$ \displaystyle \text{Price}\,\,\text{with}\,\,\text{tax}=\,\,\text{original}\,\,\text{price}+\,\,\left( {\text{original}\,\,\text{price}\,\,\times \,\,\left. {\frac{{\text{tax}\,\,\text{percentage}}}{{100}}} \right)} \right.$

$ \displaystyle \$50+\left( {\$50\,\,\times \,\,\left. {\frac{{8.25}}{{100}}} \right)} \right.\,\,=\,\,\$50+\left( {\$50\times \left. {.0825} \right)} \right.\,\,=\,\,\$50+\$4.125=\$54.125=\$54.13$

Another way we can figure percent increase is to just multiply the original amount by 1 (to make sure we include it) and also multiply it by the tax rate and add them together (this is actually using something called distributing, which we’ll talk about in Algebra):

$ \displaystyle \text{Price}\,\,\text{with}\,\,\text{tax}\,\,\text{=}\,\,\text{original}\,\,\text{price}\,\,\times \,\,\left( {1+\left. {\frac{{\text{tax}\,\,\text{percentage}}}{{100}}} \right)} \right.$

$ \displaystyle \$50\,\times \,\left( {1+\left. {\frac{{8.25}}{{100}}} \right)} \right.=\$50\,\times \,\left( {1+\left. {.0825} \right)} \right.=\$50\times 1.0825=\$54.125=\$54.13$

If we need to figure out the actual percent decrease or increase ( percent change ), we can use the following formula:

$ \displaystyle \text{Percent Increase}=\frac{{\text{New Price}-\text{Old Price}}}{{\text{Old Price}}}\,\times 100$

$ \displaystyle \text{Percent Decrease}\,=\frac{{\text{Old Price}-\text{New Price}}}{{\text{Old Price}}}\,\,\times \,100$

For example, say we want to work backwards to get the percentage of sales tax that we pay (percent increase). If we know that the original (old) price is $50 , and the price we pay (new price) is $54.13 , we could get the % we pay in tax this way  (note that since we rounded to get the 54.13 , our answer is off a little):

$ \displaystyle \text{Percent Increase (Tax)}\,\,=\frac{{54.13-50}}{{50}}\,\,\times \,100\,=\,8.26%$

Sometimes we have to work a little backwards in the problem to get the right answer. For example, we may have a problem that says something like this:

Your favorite pair of shoes are on sale for 30% off. The sale price is $62.30 . What was the original price?

To do this problem, we have to think about the fact that if the shoes are on sale for 30% , we need to pay 70% for them. Also remember that “of = times”. We can set it up this way:

$ \displaystyle \,\,.7\,\,\times \,\,?=\$62.30\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,? = \frac{{62.30}}{{.7}} = \$89$

The original price of the shoes would have been $89 before tax.

In the Algebra sections, we will address solving the following types of percentage problems, but I’ll briefly address them here if you need to do them now. If you don’t totally follow how to get the answers, don’t worry about it, since we’ll cover “word problems” later!

One other way to address percentages is the “$ \displaystyle \frac{{\text{is}}}{{\text{of}}}$” trick, which we’ll address below.

Ratios  are just a comparison of two numbers. They look a little scary since they involve fractions, but they really aren’t bad at all. Again, they are typically used when you are comparing two things — like cost of one pair of shoes to another pair, or maybe even the number of shirts you have compared to the number of jeans you have.

Let’s use that as an example. Let’s say you have about 5 shirts for every one pair of jeans you have, and you figure this same ratiois pretty typical among your friends. You can write your ratio in a fraction like $ \displaystyle \frac{5}{1}$, or you can use a colon in between the two numbers, like  5 : 1 (spoken as “ 5 to 1 ”). The fractions over 1 is actually a rate  (this word is related to the word ratio!), for example, just like when you think of miles per hour. Our rate is shirts per one pair of jeans – 5 shirts for every pair of jeans.

Also note that this particular ratio is a unit rate , since the second number (denominator in the fraction) is 1 .

Let’s say you know your friend Alicia has 7 pairs of jeans and you’re wondering how many shirts she has, based on the ratio or rate of 5 shirts to one pair of jeans.  We can do this with math quite easily by setting up the following proportion , which is an equation (setting two things equal to one another) with a ratio on each side:

How do we figure out how many shirts Alicia has? One way is just to think about reducing or expanding fractions. Let’s expand the fraction $ \displaystyle \frac{5}{1}$ to another fraction that has 7 on the bottom:

$ \displaystyle \frac{{\text{shirts}}}{{\text{jeans}}}\,=\,\frac{5}{1}\,=\,\frac{5}{1}\,\times \,1\,=\frac{5}{1}\,\,\times \,\,\frac{7}{7}\,=\frac{{35}}{7}$

Alicia would have about 35 shirts.

Now I’m going to also show you a concept called cross-multiplying , which is very, very useful, even when we get into Algebra, Geometry and up through Calculus! This is a much easier way to do these types of problems.

Remember the “butterfly up” concept when we’re comparing fractions, and remember how the fractions are equal when the “butterfly up” products are equal?

We’re going to use this concept to set the fractions or ratios equal so we know how many shirts Alicia has:

We know that 5 × 7 = 35 , so we need to know what multiplied by 1 will give us 35 .  35 !! Alicia has 35 shirts!!! See how easy that was?  Now if we didn’t have the 1 as a factor to get to 35 , we’d have to divide 35 by the number under the 5 to get the answer.  This is because dividing “undoes” multiplying.

One of my students also suggested to use the “ WON ” method for proportions. To do this, you set up a table with WON at the top.  “W” stands for Words , “O” stands for Original or Old , and “N” stands for New (in this example, for Alicia). Put the words and numbers in the table, and then cross multiply like we did earlier. Again, we get that Alicia has 35 shirts , based on my proportion of 5 shirts to every pair of jeans, and the fact that she has 7 pairs of jeans.

Let’s try a cooking example with proportions, since sometimes the recipe might give you the amounts in tablespoons, for example, and you only have a measuring spoon with teaspoons. We know from the Fractions section that 1 tablespoon = 3 teaspoons, and let’s say the recipe calls for 2 tablespoons. This seems pretty easy to do without the proportion, but let’s set it up anyway, so you can see how easy it is to use proportions:

$ \require{cancel} \displaystyle \frac{{\text{teaspoons}}}{{\text{tablespoons}}}\,\,\,\,\,={}^{6}{{\xcancel{{\frac{3}{1}\,\,\,=\,\,\,\frac{?}{2}}}}^{6}}$

Now let’s go on to a more complicated example that relates back to converting numbers back and forth between the metric system and our customary system here.  (For more discussion on the metric system, see the  Metric System  section).

Let’s say we have 13 meters of something and we want to know how many feet this is.  We can either look up how many feet are in 1 meter, or how many meters are in 1 foot – it really doesn’t matter – but we need a conversion number.

We find that 1 meter equals approximately 3.28 feet. Let’s set all this up in a proportion. Remember to keep the same unit either on the tops of the proportion, or on the sides – it works both ways:

$ \displaystyle \frac{{\text{meters}}}{{\text{meters}}}\,\,=\,\,\frac{{\text{feet}}}{{\text{feet}}}\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\frac{{\text{meters}}}{{\text{feet}}}\,\,=\,\,\frac{{\text{meters}}}{{\text{feet}}}\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\frac{{\text{feet}}}{{\text{meters}}}\,\,=\,\,\frac{{\text{feet}}}{{\text{meters}}}$

Let’s solve both two different ways to get the number of feet in 13 meters. Notice that we can turn proportions sideways, move the “ = ” sideways too, and solve – this is sort of how we got from the first equation to the second above.

Here’s an example where we have to do some dividing with our cross multiplying. Try to really understand why we have to divide by 2 to get the answer (it “undoes” the multiplying):

$ \displaystyle \frac{5}{2}\,=\,\frac{?}{9}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,5\,\,\times \,\,9\,=2\,\times \,?\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,?\,=\,\frac{{5\,\times \,9}}{2}\,=\,\frac{{45}}{2}\,=\,\,22\frac{1}{2}$

We can also use what we call unit multipliers  to change numbers from one unit to another.  The idea is to multiply fractions to get rid of the units we don’t want. You probably will use this technique some day when you take Chemistry; it may be called Dimensional Analysis .

Let’s say we want to use two unit multipliers to convert 58 inches to yards.

Since we have inches and we want to end up with yards, we’ll multiply by ratios (fractions) that relate the units to each other. We can do this because we are really multiplying by “ 1 ”, since the top and bottom amounts will be the same (just the units will be different). Let’s first set this up with the units we have to see what we’ll need to have on the top and the bottom. I put 1 ’s under the first and last items to make them look like fractions:

$ \displaystyle \frac{{58\text{ inches}}}{1}\,\,\times \,\,\frac{?}{?}\,\,\times \,\,\frac{?}{?}\,\,=\,\,\frac{{?\text{ yards}}}{1}$

We need to get rid of the inches unit on the top and somehow get the yards unit on the top; since the problem calls for 2 unit multipliers, we’ll include feet to do this:

$ \require{cancel} \displaystyle \frac{{58\text{ }\cancel{{\text{inches}}}}}{1}\,\times \,\frac{{?\text{ }\cancel{{\text{feet}}}}}{{?\text{ }\cancel{{\text{inches}}}}}\,\times \,\frac{{?\text{ }\,\text{yards}}}{{?\text{ }\cancel{{\text{feet}}}}}\,=\,\frac{{\text{? }\,\text{yards}}}{\text{1}}$

Now just fill in how many inches are in a foot, and how many feet are in a yard, and we can get the answer with real numbers:

$ \displaystyle \frac{{58\text{ }\cancel{{\text{inches}}}}}{1}\,\times \,\frac{{1\text{ }\cancel{{\text{foot}}}}}{{12\text{ }\cancel{{\text{inches}}}}}\,\times \,\frac{{1\text{ yard}}}{{3\text{ }\cancel{{\text{feet}}}}}\,=\,\frac{{58\times 1\times 1\text{ yards}}}{{1\times 12\times 3}}\,=\,\frac{{58}}{{36}}\text{ }\,\text{yards}\,=\,\frac{{29}}{{18}}\text{ }\,\text{yards}$

Here’s another example where we use two unit multipliers since we are dealing with square units:

Use two unit multipliers to convert 100 square kilometers to square meters.

$ \displaystyle \frac{{100\text{ }\cancel{{\text{kilometers}}}\times \cancel{{\text{kilometers}}}}}{1}\,\times \,\frac{{1000\text{ meters}}}{{1\text{ }\cancel{{\text{kilometer}}}}}\,\times \,\frac{{1000\text{ meters}}}{{1\text{ }\cancel{{\text{kilometer}}}}}\,=\,100,000,000\,\, \text{meter}{{\text{s}}^{2}}$

Now let’s revisit percentages and show how proportions can help with them too! One trick to use is the $ \displaystyle \frac{{\text{is}}}{{\text{of}}}$ and $ \displaystyle \frac{{\text{part}}}{{\text{whole}}}$ tricks. You can remember these since the word that comes first in the alphabet (“is” and “part”) are on the top of the fractions.

You can typically solve percentage problems by using the following formula:

$ \displaystyle \frac{{\text{is}}}{{\text{of}}}=\frac{\text{ }\!\!\%\!\!\text{ }}{{100}}$

What this means is that the number around the “is” in an equation is on top of the proportion, and the number that comes after the “of” in an equation is on bottom of the proportion, and the percentage is over the 100 .

You can also think of this as the following, but you have to remember that sometimes the part may be actually be bigger than the whole (if the percentage is greater than 100):

$ \displaystyle \frac{{\text{part}}}{{\text{whole}}}=\frac{\text{ }\!\!\%\!\!\text{ }}{{100}}$

Here are some examples, using the same problems that we did above in the Percentages section.  (Later, in the Algebra section, we’ll learn how to translate math word problems like these word-for-word from English to math.)

• What is 20% of 100 ?   Since we know that the 20 of the % part, we put that over the 100. The 100 comes after the “of”, so we put that on the bottom. Also, we’re looking for the “part” of the “whole” here.

$ \displaystyle \frac{{\text{is}}}{{\text{of}}}=\,\frac{\%}{{100}}\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\frac{{\text{part}}}{{\text{whole}}}=\frac{\%}{{100}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{?}{{100}}=\frac{{20}}{{100}}\,\,\,\,\,\,\,\,\,?=20$

• 100 is what percentage of 200 ?   The 100 is close to the “is” so we put that on the top. The 200 comes after the “of”, so we put that on the bottom. Also, we know the 100 is the “part” of the 200 .

$ \displaystyle \frac{{\text{is}}}{{\text{of}}}=\frac{\%}{{100}}\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\frac{{\text{part}}}{{\text{whole}}}=\frac{\%}{{100}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{100}}{{200}}=\frac{{?\,\,\,\%}}{{100}}\,\,\,\,\,\,\,\,\,?=50$

• 200 is 50% of what number?    The 200 is close to the “is” and we don’t know what the “of” is. The 50 is the percentage. Also, 200 is the “part”, so we need to find the “whole”.

$ \displaystyle \frac{{\text{is}}}{{\text{of}}}=\frac{%}{{100}}\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\frac{{\text{part}}}{{\text{whole}}}=\frac{%}{{100}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{200}}{?}=\frac{{50}}{{100}}\,\,\,\,\,\,\,\,\,?=400$

Here are a few more problems on rates and percentages :

Remember also – if you’re not quite sure what you’re doing, think of the problems with easier numbers and see how you’re doing it!  This can help a lot of the time. Learn these rules and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the  Mathway  site, where you can register for the  full version  (steps included) of the software.  You can even get math worksheets.

You can also go to the  Mathway  site here , where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Negative Numbers and Absolute Value   – you are ready!!

Combining Ratios

What is a Combining Ratio question ?

A combining ratio question is when you are given two ratios that have a common variable and are asked to combine them into one overall ratio.

For instance, you know the ratios a:b and b:c and you are then asked to find the ratio a:b:c .

The way to solve these is to find a common multiple of the part that appears in both ratios.

In the example shown, a appears in both ratios so we find the lowest common multiple of 3 and 4 (which is 12). We multiply all parts of the ratio a:b by 4 and all parts of the ratio a:c by 3.

This gives the overall ratio a:b:c

Combining Ratio Question (Edexcel Higher , Specimen Set 2, Paper 2 , Question 6 )

Example Question

the number of cows and the number of sheep are in the ratio 6 : 5 

the number of sheep and the number of pigs are in the ratio 2 : 1 

The total number of cows, sheep and pigs on the farm is 189 

How many sheep are there on the farm?

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