Question
solve ratio problems with proportions practice
Answers
GPT-4o mini
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
1. **Ratio**: A ratio is a comparison between two quantities. It can be written in several ways: `3:4`, `3 to 4`, or `3/4`.
2. **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
1. The ratio of boys to girls in a classroom is 5:7. If there are 35 boys, how many girls are there?
2. A car travels 150 miles in 3 hours. If it keeps the same speed, how far will it travel in 5 hours?
3. The ratio of red to green marbles is 3:2. If there are 30 red marbles, how many green marbles are there?
4. If the ratio of apples to oranges in a basket is 4:5 and there are 24 apples, how many oranges are there?
5. 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
1. **Solution**: Set up the proportion \(\frac{5}{7} = \frac{35}{x}\) → \(5x = 7 \cdot 35\) → \(5x = 245\) → \(x = 49\). There are **49 girls**.
2. **Solution**: Speed = distance/time = \(150/3 = 50\) mph. In 5 hours, distance = speed × time = \(50 \times 5 = 250\) miles. Will travel **250 miles**.
3. **Solution**: Set up the proportion \(\frac{3}{2} = \frac{30}{x}\) → \(3x = 2 \cdot 30\) → \(3x = 60\) → \(x = 20\). There are **20 green marbles**.
4. **Solution**: Set up the proportion \(\frac{4}{5} = \frac{24}{x}\) → \(4x = 5 \cdot 24\) → \(4x = 120\) → \(x = 30\). There are **30 oranges**.
5. **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!
### Understanding Ratios and Proportions
1. **Ratio**: A ratio is a comparison between two quantities. It can be written in several ways: `3:4`, `3 to 4`, or `3/4`.
2. **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
1. The ratio of boys to girls in a classroom is 5:7. If there are 35 boys, how many girls are there?
2. A car travels 150 miles in 3 hours. If it keeps the same speed, how far will it travel in 5 hours?
3. The ratio of red to green marbles is 3:2. If there are 30 red marbles, how many green marbles are there?
4. If the ratio of apples to oranges in a basket is 4:5 and there are 24 apples, how many oranges are there?
5. 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
1. **Solution**: Set up the proportion \(\frac{5}{7} = \frac{35}{x}\) → \(5x = 7 \cdot 35\) → \(5x = 245\) → \(x = 49\). There are **49 girls**.
2. **Solution**: Speed = distance/time = \(150/3 = 50\) mph. In 5 hours, distance = speed × time = \(50 \times 5 = 250\) miles. Will travel **250 miles**.
3. **Solution**: Set up the proportion \(\frac{3}{2} = \frac{30}{x}\) → \(3x = 2 \cdot 30\) → \(3x = 60\) → \(x = 20\). There are **20 green marbles**.
4. **Solution**: Set up the proportion \(\frac{4}{5} = \frac{24}{x}\) → \(4x = 5 \cdot 24\) → \(4x = 120\) → \(x = 30\). There are **30 oranges**.
5. **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!