Ratio, proportion & rates of change
Ratio: Sharing, Fractions & Combining
Year 10 · Year 11
- ✓By the end of this lesson students will be able to understand and use ratio notation.
- ✓By the end of this lesson students will be able to simplify ratios to their simplest form.
- ✓By the end of this lesson students will be able to divide a given quantity into two or more parts in a given ratio.
- ✓By the end of this lesson students will be able to convert between ratios and fractions.
- ✓By the end of this lesson students will be able to solve problems involving combining two or more ratios.
Key concepts
A ratio compares the sizes of two or more quantities. It shows how much of one quantity there is compared to another. Ratios are often used to compare parts of a whole or to compare different quantities. The colon (:) is used to separate the quantities. For example, 2:3 means for every 2 parts of the first quantity, there are 3 parts of the second. Ratios do not have units.
Ratios can be simplified by dividing all parts by their highest common factor (HCF), similar to simplifying fractions. A ratio is in its simplest form when the parts are whole numbers and have no common factors other than 1. For example, the ratio 10:15 can be simplified by dividing both numbers by 5, resulting in 2:3.
To share a total quantity in a given ratio, first find the total number of parts in the ratio. Then, divide the total quantity by the total number of parts to find the value of one part. Finally, multiply the value of one part by each number in the ratio to find the share for each person or category.
A ratio can be directly related to fractions. If a ratio is a:b, then the total number of parts is a + b. The first quantity represents a/(a+b) of the total, and the second quantity represents b/(a+b) of the total. Conversely, if a quantity is a fraction of another, this can be expressed as a ratio.
When two ratios share a common element, they can be combined into a single, extended ratio. To do this, find a common multiple for the part representing the common element in both ratios. Adjust both ratios so that the common element has the same value, then combine the other parts to form the new, combined ratio.
Key facts to remember
- 1A ratio compares the sizes of two or more quantities.
- 2Ratios are written using a colon, e.g., a:b, and do not have units.
- 3Ratios can be simplified by dividing all parts by their highest common factor (HCF).
- 4To share a quantity in a ratio, find the total number of parts, then the value of one part.
- 5If a ratio is a:b, the first quantity is a/(a+b) of the total, and the second is b/(a+b) of the total.
- 6When combining ratios, make the common element the same value in both ratios by finding a common multiple.
Worked examples
Example 1
Share £72 in the ratio 3:5.
Answer
£27 and £45.
Always check your answer by adding the individual shares: £27 + £45 = £72, which matches the original total.
Example 2
In a class, the ratio of boys to girls is 2:3. If there are 30 students in total, how many girls are there? What fraction of the class are boys?
Answer
There are 18 girls. 2/5 of the class are boys.
Ensure you answer all parts of the question. The fraction represents 'part over whole'.
Example 3
The ratio of apples to bananas in a fruit bowl is 2:3. The ratio of bananas to oranges is 6:5. Find the ratio of apples to bananas to oranges.
Answer
4:6:5
Always ensure the combined ratio is in its simplest form. In this case, 4, 6, and 5 have no common factors other than 1.
Common mistakes
- ✗Not simplifying ratios to their simplest form (e.g., leaving 10:15 instead of 2:3).
- ✗Dividing by one of the ratio numbers instead of the sum of the parts when sharing a quantity.
- ✗Confusing which part of the ratio corresponds to which quantity (e.g., assigning the first number to the second item mentioned).
- ✗Incorrectly converting between ratios and fractions (e.g., thinking a ratio of 1:2 means 1/2 of the total, instead of 1/3).
- ✗Failing to multiply all parts of a ratio when adjusting it to combine with another ratio.
Exam tips
- ★Always read the question carefully to identify what quantities are being compared and what the total amount is (if sharing).
- ★Show all your working steps, especially when sharing a quantity, as method marks are often awarded.
- ★For sharing problems, check your answer by adding up the individual shares to ensure they total the original amount.
- ★When combining ratios, clearly write out the adjusted ratios before combining them to avoid errors, and ensure the final ratio is in its simplest form.
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