Ratio, proportion and rates of change Revision Notes

What is a Ratio?

A ratio compares the relative sizes of two or more quantities. It tells you how much of one thing there is compared to another. Ratios can be written using a colon (e.g., 2:3) or as a fraction (e.g., 2/3), although the fraction form is more common when comparing two parts to a whole. For example, if you have 2 red balls and 3 blue balls, the ratio of red to blue balls is 2:3.

What are Equivalent Ratios?

Equivalent ratios are ratios that express the same relationship between quantities. They might look different, but they represent the same proportion. Just like equivalent fractions (e.g., 1/2 = 2/4), equivalent ratios maintain the same relative sizes. For instance, a ratio of 1:2 is equivalent to 2:4, 3:6, and 10:20. These all mean that for every one unit of the first quantity, there are two units of the second quantity.

How to Find Equivalent Ratios

To find an equivalent ratio, you must multiply or divide all parts of the ratio by the same non-zero number. If you have a ratio a:b, and you multiply both 'a' and 'b' by a number 'k', you get the equivalent ratio (a × k) : (b × k). Similarly, if you divide both 'a' and 'b' by a common factor 'c', you get the equivalent ratio (a ÷ c) : (b ÷ c). This process ensures the proportional relationship remains unchanged. For example, to find an equivalent ratio to 3:5, you could multiply both by 2 to get 6:10, or by 10 to get 30:50. All these ratios are equivalent.

Simplifying Ratios

Simplifying a ratio means writing it in its simplest form, where the numbers are as small as possible. This is achieved by dividing all parts of the ratio by their greatest common factor (GCF). For example, to simplify the ratio 18:24, find the GCF of 18 and 24, which is 6. Divide both parts by 6: (18 ÷ 6) : (24 ÷ 6) = 3:4. The ratio 3:4 is the simplified form of 18:24, and they are equivalent ratios.

What is a Ratio?

A ratio compares how much there is of one thing compared to another. For instance, a ratio of 1:2 means for every 1 unit of the first item, there are 2 units of the second item. Ratios can involve two or more numbers, e.g., 2:3:5. The order of the numbers in a ratio is important as it refers to the corresponding items in that order. Ratios do not have units, but the quantities being compared should be in the same units when setting up or using the ratio for division.

Steps for Dividing a Quantity in a Ratio

To divide a total quantity into specific shares according to a given ratio, follow these steps: First, add all the numbers in the ratio to find the 'total number of parts'. Second, divide the total quantity by the 'total number of parts' to find the 'value of one part'. Third, multiply the 'value of one part' by each number in the original ratio to find each individual share. Fourth, check your work by adding all the individual shares; their sum should equal the original total quantity. For example, to divide £60 in the ratio 1:2:3:

1. Total parts: 1 + 2 + 3 = 6 parts.
2. Value of one part: £60 ÷ 6 = £10 per part.
3. Individual shares: 1 × £10 = £10, 2 × £10 = £20, 3 × £10 = £30.
4. Check: £10 + £20 + £30 = £60.

Working with Ratios and Totals

Sometimes, you may be given a part of the total and asked to find another part or the total itself. If, for example, person A gets £15 and the ratio of A:B is 3:5, then 3 parts correspond to £15. The value of one part is £15 ÷ 3 = £5. Then, person B gets 5 parts, so 5 × £5 = £25. The total amount is £15 + £25 = £40. Always identify what the given number represents (a total or a specific part) before performing calculations.

Simplifying Ratios

Before dividing a quantity, it's often helpful to simplify the ratio, if possible, by dividing all parts of the ratio by their highest common factor. For example, the ratio 6:9 can be simplified to 2:3 by dividing both numbers by 3. Dividing a quantity using the simplified ratio will result in the same proportional shares. However, ensure the problem doesn't require using the original form for some specific reason.

Understanding Ratios

A ratio compares the sizes of two or more quantities. For example, a ratio of 2:3 means there are two parts of the first quantity for every three parts of the second quantity. Ratios can be used to describe distributions, proportions, or scales. It's important to remember that the order of the numbers in a ratio matters – 2:3 is different from 3:2. The numbers in a ratio do not necessarily represent the actual quantities, but rather the relative sizes of the parts.

Steps for Dividing a Quantity in a Ratio

1. Find the total number of parts: Add all the numbers in the given ratio. For example, if the ratio is A:B, the total parts are A + B. If the ratio is A:B:C, the total parts are A + B + C.
2. Calculate the value of one part: Divide the total quantity to be shared by the total number of parts you found in step 1.
3. Determine each share: Multiply the value of one part (from step 2) by each number in the original ratio. This will give you the amount for each corresponding 'share'.

Checking Your Answer

After you have calculated all the individual shares, it's a good practice to add them all together. The sum of all the calculated shares should be equal to the original total quantity that was divided. This helps confirm that your calculations are correct and that no errors were made during the division process. If the sum does not match the original quantity, recheck your steps.

Applying Ratio Division in Context

Dividing quantities in a given ratio is a fundamental skill used in various real-world scenarios. This includes sharing money fairly, mixing ingredients in recipes according to specific proportions, scaling up or down measurements in designs, or allocating resources. Understanding the steps allows for practical application in problem-solving questions. Always ensure you identify the total quantity and the correct ratio before starting calculations.

What is a Ratio?

A ratio compares the sizes of two or more quantities. It shows how much of one quantity there is compared to another. Ratios can be written in several ways, such as 'A to B', 'A:B', or as a fraction A/B. For example, if an orange drink is made with 1 part concentrate and 4 parts water, the ratio of concentrate to water is 1:4. This indicates that for every 1 unit of concentrate, there are 4 units of water.

Simplifying Ratios

Just like fractions, ratios can often be simplified. To simplify a ratio, divide all parts of the ratio by their greatest common factor (GCF). For instance, a ratio of 10:15 can be simplified by dividing both numbers by 5, resulting in the simpler ratio 2:3. Similarly, a ratio of 12:18:24 can be simplified by dividing all parts by 6, giving 2:3:4. Simplifying ratios makes them easier to understand and compare.

Understanding Fractions

A fraction represents a part of a whole. It consists of a numerator (the number above the line) and a denominator (the number below the line). The denominator shows the total number of equal parts the whole is divided into, while the numerator shows how many of those parts are being considered. For example, 3/4 means three out of four equal parts. Equivalent fractions represent the same value even though they have different numerators and denominators (e.g., 1/2 and 2/4).

Ratios and Fractions Relationship

There is a direct link between ratios and fractions. If a total is split into parts in a given ratio, say A:B, then the total number of 'parts' is A+B. The fraction of the total represented by A is A/(A+B), and the fraction represented by B is B/(A+B). For example, if the ratio of boys to girls in a class is 2:3, the total number of parts is 2+3=5. The fraction of boys in the class is 2/5, and the fraction of girls is 3/5.