Content

• To step
• Part 1 of 2: Noting a ratio
• Part 2 of 2: Using proportions in math problems
• Example exercises
• Tips
• Necessities

Proportions or ratios are mathematical expressions that compare two or more numbers. Ratios can compare fixed quantities and numbers or can be used to compare parts of the whole. Ratios can be calculated and noted in different ways, but the principles are the same for all ratios. To get started with ratios, see Step 1 below.

To step

Part 1 of 2: Noting a ratio

1. Understand how proportions are used. You encounter relationships everywhere, in the scientific world or at home. The simplest ratios compare only two values, but more is also possible.
   • An example: in a class with 20 students, of which 5 girls and 15 boys, we can express the number of girls and boys as a ratio.
2. Write a ratio with a colon. A common way to indicate a ratio is with a colon between the numbers. If you compare two numbers, you write it down for example as 7: 13 and there are 3 or more numbers, for example as follows 10: 2: 23.
   • So in our classroom we can write the ratio girls to boys as follows: 5 girls: 15 boys. Optionally, you can omit the indication, as long as you remember what the ratio stands for.
3. A ratio is the same as a fraction, so it can be simplified. You do this by dividing all terms of the ratio by the common denominators, until there are no common denominators left.But when you do this, it is important not to forget the original numbers of the ratio. See below.
   • In the classroom example, there were 5 girls and 15 boys. Both sides of the ratio are divisible by 5. This allows you to simplify the ratio to 1 girl: 3 boys.
     • But we shouldn't lose sight of the original numbers. There are not 4 but 20 students in total in the class. The simplified ratio only compares the relationship between the number of boys and girls. There are 3 boys to 1 girl in the relationship or fraction, not 3 boys and 1 girl in the class.
   • Some relationships cannot be simplified. For example, 3:56 cannot be simplified because the 2 numbers do not have equal factors - 3 is prime and 56 is not divisible by 3.
4. There are also alternative methods of writing down ratios. While the colon for noting ratios may be the easiest, there are other ways too, without making any difference to the ratio. See below:
   • Ratios can also be displayed as "3 to 6" or "11 to 4 to 20".
   • You can also write proportions as a fraction. Often times using both terms leads to some confusion, but fractions are proportions and vice versa. You can therefore also write a ratio with a division line. For example the ratio 3/5 and the fracture 3/5 do not differ from each other. As with the example of the class: there were 3 boys to each girl, a ratio of 1: 3, but as a fraction this expresses the same thing, namely 1/3 of the total number of students is a girl.

Part 2 of 2: Using proportions in math problems

1. Use multiplication or division to change ratios without changing the ratio. By multiplying or dividing both terms of a ratio by a certain number, the same ratio is obtained, but with larger or smaller numbers.
   • For example, suppose you are a teacher and you are asked to make the class 5 times the size, but with the same ratio of boys and girls. If there are now 8 girls and 11 boys in the class, how many are in the new class? Read on for the solution:
     • 8 girls and 11 boys, so a ratio of 8 : 11. This ratio therefore indicates that regardless of the size of the class, there are 8 girls to 11 boys.
     • (8 : 11) × 5
     • (8 × 5 : 11 × 5)
     • (40:55). The new class consists of 40 girls and 55 guys - 95 students in total!
2. Use cross multiplication to find the unknown variable when working with two equivalent ratios. Another known problem is the one where you are asked to calculate the unknown of a ratio. Cross multiplication makes working out this very easy. Write each ratio as a fraction, make them equal, and then cross multiply to solve.
   • As an example, suppose we have a group of students of 2 boys and 5 girls. If we want to keep the ratio intact, how many boys are there in a group of 20 girls? To solve this we make two ratios, one of which with the unknown variable: 2 boys: 5 girls = x boys: 20 girls. In fractional form it looks like this: 2/5 = x / 20. To solve this, use cross multiplication. See below:
     • 2/5 = x / 20
     • 5 × x = 2 × 20
     • 5x = 40
     • x = 40/5 = 8. So there are 20 girls and 8 guys.
3. Use ratios to find unknown quantities, where a different one is given. If you are dealing with a variable that determines the relationship between different quantities, of which 1 or more are unknown, you can find the value of each unknown, using only one known quantity. Often times, these types of statements involve calculating the amounts of ingredients in a recipe. To determine the unknown quantities, divide the known term of the ratio by the given quantity; share after that any term in the relationship by the answer you get. An example will make it all clearer:
   • Suppose our class is baking cookies as an assignment. If the dough recipe consists of flour, water and butter in the ratio 20: 8: 4, and each student gets 5 cups of flour; how much water and butter does each student need? To solve this, first divide the term of the ratio that corresponds to the known ratio (20) by the known amount (5 cups). Then divide each term in the ratio by the answer you get to find the exact amount for each. See below:
     • 20 / 5 = 4
     • 20/4 : 8/4 : 4/4
     • 5: 2: 1. So, 5 cups of flour, 2 cups of water and 1 cup of butter.

Example exercises

• Biscuits are made from butter and sugar in a ratio of 5: 3. If 7 parts of butter are used, how much sugar is needed?
  • To do this, use the ratio in the form of a fraction. In this case, we'll turn it into a decimal - about 1.67.
  • The formula is now ready to use. We want to find the amount of sugar, so we leave it for what it is and calculate the fraction of butter / 1.67, so 7 / 1.67 = 4.192
• The part about proportions is proportional sharing. When a total quantity is divided into pieces, a ratio is created. For example: Annemiek, Anna and Anton all work in their mother's shop. Annemiek worked an hour, Anna 3 and Anton 6 hours (so a ratio 1: 3: 6). Mother gives them a total amount and asks them to divide this themselves in the correct proportion. The total amount was € 100. You do this by adding up the parts of the ratio so you know how much each part is worth. 1: 3: 6 then becomes 1 + 3 + 6 = 10 so € 100/10 = € 10 so we now know that each part of the ratio is worth € 10 ... and therefore everyone gets a wage of € 10 per hour. Now we can use this to calculate what each person has earned. Annemiek will receive € 10, Anna will receive € 30 and Anton will receive € 60. Check this by adding up all the wages, which should then amount to € 100. 10 + 30 + 60 = 100. Correct!

Tips

• Simplify proportions using the ab / c button on your calculator (this is for writing mixed fractions and simplifying). For example, if you have 8:12, you enter "8 ab / c 12" = and you get 2/3, which means the ratio 2: 3.

Necessities

• Calculator (optional)