Ratio, proportion and rates of change Revision Notes

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:b, a to b, or a/b. For instance, if there are 5 apples and 3 oranges, the ratio of apples to oranges is 5:3. Ratios should always be expressed in their simplest form, similar to fractions. To simplify a ratio, divide all parts of the ratio by their highest common factor. For example, the ratio 10:15 simplifies to 2:3 by dividing both by 5.

Sharing in a Given Ratio

To share a total quantity in a given ratio, follow these steps: 1. Add the parts of the ratio together to find the 'total parts'. 2. Divide the total quantity by the 'total parts' to find the value of 'one part'. 3. Multiply the value of 'one part' by each number in the ratio to find the amount for each share. For example, to share £60 in the ratio 2:3:

1. Total parts = 2 + 3 = 5
2. Value of one part = £60 / 5 = £12
3. Shares are 2 * £12 = £24 and 3 * £12 = £36. (Check: £24 + £36 = £60).

Understanding Proportion

Proportion describes the relationship between two ratios that are equal. If two quantities are in proportion, it means that as one quantity changes, the other quantity changes in a consistent way. Direct proportion means that as one quantity increases, the other quantity increases at the same rate, and their ratio remains constant. Inverse proportion means that as one quantity increases, the other decreases. When solving direct proportion problems, a common method is to find the value of a single unit before scaling up. For example, if 4 books cost £10, then 1 book costs £10/4 = £2.50. So 6 books would cost 6 * £2.50 = £15.

Solving Ratio Problems using Unitary Method

The unitary method is an effective way to solve many ratio and proportion problems. It involves finding the value of a single unit or part, then using that to calculate other values. For example, if a recipe calls for 2 cups of flour for 8 cookies:

1. Find the amount of flour for 1 cookie: 2 cups / 8 cookies = 0.25 cups per cookie.
2. If you want to make 20 cookies, multiply the unit rate by the desired number: 0.25 cups/cookie * 20 cookies = 5 cups of flour. This method simplifies complex scaling and comparisons by breaking down the problem into a per-unit basis.

What is Direct Proportion?

Direct proportion describes a relationship between two quantities where one is a constant multiple of the other. If quantity 'A' is directly proportional to quantity 'B', then as 'A' increases, 'B' increases by the same factor, and similarly if 'A' decreases, 'B' decreases by the same factor. This means the ratio of the quantities remains constant. For example, if the number of hours worked increases, the amount earned often increases directly proportionally.

The Proportionality Statement and Equation

We write 'y is directly proportional to x' as $y \propto x$. To convert this proportionality into an equation, we introduce a 'constant of proportionality', 'k'. The equation becomes $y = kx$. This constant 'k' is a fixed value that links 'y' and 'x'. It can be found by dividing any paired value of 'y' by 'x' ($k = y/x$), as long as 'x' is not zero. Remember that 'k' must be a non-zero constant.

Finding the Constant of Proportionality (k)

To find 'k', you need at least one pair of corresponding values for 'x' and 'y'. Substitute these values into the equation $y = kx$. For example, if $y$ is directly proportional to $x$, and $y = 10$ when $x = 2$, then $10 = k \times 2$. Solving for $k$, we get $k = 10/2 = 5$. Once 'k' is found, you have the full relationship $y = 5x$ and can use it to find other unknown values.

Using the Relationship to Solve Problems

Once you have determined the equation $y = kx$ by finding 'k', you can use it to predict values. For instance, if $y = 5x$ (from the example above), you can find $y$ when $x = 7$ by calculating $y = 5 \times 7 = 35$. Likewise, you can find $x$ when $y = 45$ by solving $45 = 5x$, which gives $x = 45/5 = 9$. This method allows you to solve for any unknown quantity in the direct proportionality relationship.

Core method

Two quantities are in direct proportion when multiplying one by a constant multiplies the other by the same constant. The relationship can be written y = kx, where k is the constant of proportionality. Find k from a known pair, then use it to calculate missing values. The essential approach is to identify the mathematical structure before calculating, retain exact values where possible, and show enough working for each conclusion to be checked independently.

Worked reasoning and checks

If 5 notebooks cost £12, the cost per notebook is k = 12 ÷ 5 = £2.40. Therefore C = 2.4n, and 8 notebooks cost 2.4 × 8 = £19.20. A direct-proportion graph is a straight line through the origin because zero of one quantity corresponds to zero of the other. Check the final result against the original information, including signs, units, scale and whether the requested form or degree of accuracy has been supplied.

Core method

This is Higher-tier content, so solutions should communicate multi-step reasoning precisely. Higher direct-proportion problems may involve powers, roots or several variables. Translate the wording accurately: y directly proportional to x² means y = kx², while y directly proportional to √x means y = k√x. Determine k from known values before solving or comparing cases. The essential approach is to identify the mathematical structure before calculating, retain exact values where possible, and show enough working for each conclusion to be checked independently.

Worked reasoning and checks

If y ∝ x² and y = 45 when x = 3, then 45 = 9k, so k = 5 and y = 5x². When x = 7, y = 245. For percentage changes, if x is multiplied by a factor r, then x² and therefore y are multiplied by r². Check the final result against the original information, including signs, units, scale and whether the requested form or degree of accuracy has been supplied.