Certainly! Proportion and ratio problems often involve comparing two quantities and finding an equivalent relationship between them. Here’s a guide with examples and practice problems for you:
Understanding Ratios and Proportions
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Ratio: A ratio is a comparison between two quantities. It can be written in several ways:
3:4,3 to 4, or3/4. -
Proportion: A proportion states that two ratios are equal. It can be expressed as an equation: \(\frac{a}{b} = \frac{c}{d}\).
Solving Ratios and Proportions
To solve a proportion, you can use cross-multiplication. For example, if you have:
\[ \frac{a}{b} = \frac{c}{d} \]
You can cross-multiply:
\[ a \cdot d = b \cdot c \]
Example Problems
Example 1
If the ratio of cats to dogs in a shelter is 2:3, and there are 12 cats, how many dogs are there?
Solution:
Set up the proportion:
\[ \frac{2}{3} = \frac{12}{x} \]
Cross-multiply:
\[ 2x = 3 \cdot 12 \] \[ 2x = 36 \]
Now divide both sides by 2:
\[ x = 18 \]
So, there are 18 dogs.
Example 2
A recipe requires a ratio of 4 cups of flour to 3 cups of sugar. If you have 12 cups of flour, how much sugar do you need?
Solution:
Set up the proportion:
\[ \frac{4}{3} = \frac{12}{x} \]
Cross-multiply:
\[ 4x = 3 \cdot 12 \] \[ 4x = 36 \]
Now divide by 4:
\[ x = 9 \]
You need 9 cups of sugar.
Practice Problems
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The ratio of boys to girls in a classroom is 5:7. If there are 35 boys, how many girls are there?
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A car travels 150 miles in 3 hours. If it keeps the same speed, how far will it travel in 5 hours?
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The ratio of red to green marbles is 3:2. If there are 30 red marbles, how many green marbles are there?
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If the ratio of apples to oranges in a basket is 4:5 and there are 24 apples, how many oranges are there?
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A map has a scale of 1:100,000. If two cities are 250 km apart, how far apart are they on the map?
Solutions to Practice Problems
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Solution: Set up the proportion \(\frac{5}{7} = \frac{35}{x}\) → \(5x = 7 \cdot 35\) → \(5x = 245\) → \(x = 49\). There are 49 girls.
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Solution: Speed = distance/time = \(150/3 = 50\) mph. In 5 hours, distance = speed × time = \(50 \times 5 = 250\) miles. Will travel 250 miles.
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Solution: Set up the proportion \(\frac{3}{2} = \frac{30}{x}\) → \(3x = 2 \cdot 30\) → \(3x = 60\) → \(x = 20\). There are 20 green marbles.
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Solution: Set up the proportion \(\frac{4}{5} = \frac{24}{x}\) → \(4x = 5 \cdot 24\) → \(4x = 120\) → \(x = 30\). There are 30 oranges.
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Solution: The map scale means 1 cm represents 100,000 cm (or 1 km). The distance between cities on the map = \(250 \text{ km} \div 100 \text{ km/cm} = 2.5\) cm. They are 2.5 cm apart on the map.
Feel free to try solving the practice problems and check your work with the solutions provided! If you need further explanations or have other questions, let me know!
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