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Conversions, Ratios & Proportions
IMAT Interactive Study Tool
1. Core Theory
Ratios
A ratio is a way of comparing two or more quantities of the same kind. It shows the relative size of the quantities.
If the ratio of apples to oranges is 2:3, it means for every 2 apples, there are 3 oranges. Ratios can be simplified just like fractions (e.g., a ratio of 4:6 is the same as 2:3).
Proportions
A proportion is an equation that states that two ratios are equal. Proportions are used to solve problems where quantities scale up or down at the same rate.
If (frac{a}{b} = frac{c}{d}), then (ad = bc) (cross-multiplication).
Conversions
Conversions involve changing a quantity from one unit to another (e.g., kilometers to meters). The key is to multiply by a conversion factor, which is a fraction equal to 1.
For example, since 1 km = 1000 m, the conversion factors are (frac{1000 , m}{1 , km}) and (frac{1 , km}{1000 , m}). You choose the one that allows the original units to cancel out.
2. Concept Check
1. True or False: The ratio 5:10 is the same as the ratio 1:2.
2. True or False: If two quantities are in proportion, doubling one quantity means you must double the other.
3. Solved Examples
Example 1: Ratio Problem
A recipe for a drink requires mixing juice and water in the ratio 2:5. If you use 300 mL of juice, how much water do you need?
Step 1: Set up a proportion.
Let (w) be the amount of water needed. The ratio of juice to water must remain the same.
(frac{text{juice}}{text{water}} = frac{2}{5} = frac{300}{w})
Step 2: Solve for (w) using cross-multiplication.
(2 times w = 5 times 300 implies 2w = 1500)
(w = frac{1500}{2} = 750 , mL)
Conclusion: You need 750 mL of water.
Example 2: Unit Conversion
A car is traveling at 72 km/h. What is its speed in m/s?
Step 1: Identify the conversion factors.
1 km = 1000 m
1 hour = 3600 seconds
Step 2: Set up the calculation to cancel units.
(72 , frac{text{km}}{text{h}} times frac{1000 , text{m}}{1 , text{km}} times frac{1 , text{h}}{3600 , text{s}})
Step 3: Calculate the result.
(frac{72 times 1000}{3600} , frac{text{m}}{text{s}} = frac{72000}{3600} , frac{text{m}}{text{s}} = 20 , m/s)
Conclusion: The car’s speed is 20 m/s.
4. MCQ Practice
1. A map has a scale of 1:50,000. If two towns are 4 cm apart on the map, what is the actual distance between them in kilometers?
2. If 5 machines can produce 150 widgets in a day, how many widgets can 8 machines produce in a day, assuming they all work at the same rate?
3. A square has an area of 4 cm². If you convert this area to mm², what is the result? (1 cm = 10 mm)
4. The ratio of boys to girls in a class is 3:4. If there are 12 boys, how many students are there in total?
5. To convert a speed from meters per second (m/s) to kilometers per hour (km/h), you should: