Overview
This lecture covers solving different types of ratio problems, including combining ratios, using ratios to solve value-sharing questions, and finding proportions using ratio and fraction methods.
Combining Ratios
- To combine ratios sharing a common element, make the common part equal in both ratios by finding a common multiple.
- Example: Combine red:green = 3:2 and green:yellow = 5:2 by making green = 10, resulting in red:green:yellow = 15:10:4.
- Combine a:b = 5:2 and b:c = 3:1 by making b = 6, leading to a:b:c = 15:6:2.
Ratio Sharing Problems
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Combine a:b = 1:2 and b:c = 7:5 by making b = 14, yielding a:b:c = 7:14:10.
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If b gets 8 more than c and there are 4 more ratio parts for b, each part equals 2 (8 ÷ 4).
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Amounts: a = 7×2 = 14, b = 14×2 = 28, c = 10×2 = 20.
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To share £120 in ratio a:b:c = 5:4:6, sum parts (15) so each part = £8 (120 ÷ 15).
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Amounts: a = 40, b = 32, c = 48.
Proportions in Ratio and Fraction Form
- For adults:children = 3:2, children are 2/5 of total; if 40% are over 12, use 2/5 × 2/5 = 4/25.
- For red:green = 4:3 and red circles:red squares = 5:2, red shapes are 4/7; red circles are 5/7 of red shapes, so 5/7 × 4/7 = 20/49.
Ratios on a Line
- For points a, b, c, d on a line, with AB:BD = 2:5 and AC:CD = 7:3, adjust ratios to have equal totals to find individual segments.
- Example: Multiply AB:BD by 10 (get 20:50) and AC:CD by 7 (get 49:21) to get AB:BC:CD = 20:29:21.
- Use subtraction of overlapping segments for the missing ratio part.
Worked Examples
- Combine men:women = 3:4; 30% of women are over 30: 4/7 × 3/10 = 12/70 = 6/35.
- Combine a:b = 5:3 and b:c = 7:2 by making b = 21, giving a:b:c = 35:21:6.
Key Terms & Definitions
- Ratio — A comparison showing how many times one value contains or is contained within another.
- Proportion — A part or fraction of a whole, often derived from ratios.
- Equivalent Ratio — Ratios that express the same relationship after scaling.
Action Items / Next Steps
- Practice combining ratios with shared elements.
- Solve ratio sharing and proportion questions given in the lecture.
- Review methods for converting ratios to fractions to find proportions.