Chapter 1: What is a Ratio?
A ratio is a comparison of two or more quantities that shows their relative sizes. Ratios can describe how much of one thing there is compared to another.
- It can be written using a colon: 3:2
- As a fraction: 3/2
- Or using words: "3 to 2"
Example:
If there are 6 apples and 4 oranges in a basket, the ratio of apples to oranges is 6:4 or simplified as 3:2.
Ratios help us compare, scale, and solve real-life problems, such as recipes, maps, and more.
Chapter 2: Examples of the Topics
1. Changing Improper Fractions to Mixed Numbers
Example:
\( \frac{11}{4} = 2 \frac{3}{4} \) (11 divided by 4 is 2 with a remainder of 3)
Example:
\( \frac{11}{4} = 2 \frac{3}{4} \) (11 divided by 4 is 2 with a remainder of 3)
2. Changing Mixed Numbers to Improper Fractions
Example:
\( 2 \frac{3}{4} = \frac{11}{4} \) (2 × 4 + 3 = 11)
Example:
\( 2 \frac{3}{4} = \frac{11}{4} \) (2 × 4 + 3 = 11)
3. Adding Fractions with Like Denominators
\( \frac{2}{7} + \frac{3}{7} = \frac{5}{7} \)
\( \frac{2}{7} + \frac{3}{7} = \frac{5}{7} \)
4. Subtracting Fractions with Like Denominators
\( \frac{5}{9} - \frac{2}{9} = \frac{3}{9} = \frac{1}{3} \)
\( \frac{5}{9} - \frac{2}{9} = \frac{3}{9} = \frac{1}{3} \)
5. Adding or Subtracting Fractions with Unlike Denominators
\( \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \)
\( \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \)
6. Adding Mixed Numbers with Unlike Denominators
\( 1 \frac{1}{2} + 2 \frac{2}{3} = \frac{3}{2} + \frac{8}{3} = \frac{9}{6} + \frac{16}{6} = \frac{25}{6} = 4 \frac{1}{6} \)
\( 1 \frac{1}{2} + 2 \frac{2}{3} = \frac{3}{2} + \frac{8}{3} = \frac{9}{6} + \frac{16}{6} = \frac{25}{6} = 4 \frac{1}{6} \)
7. Subtracting Mixed Numbers with Unlike Denominators
\( 3 \frac{3}{4} - 1 \frac{2}{3} = \frac{15}{4} - \frac{5}{3} = \frac{45}{12} - \frac{20}{12} = \frac{25}{12} = 2 \frac{1}{12} \)
\( 3 \frac{3}{4} - 1 \frac{2}{3} = \frac{15}{4} - \frac{5}{3} = \frac{45}{12} - \frac{20}{12} = \frac{25}{12} = 2 \frac{1}{12} \)
8. Estimating Sums and Differences of Fractions and Mixed Numbers
\( 4 \frac{5}{6} \approx 5 \), \( 2 \frac{1}{8} \approx 2 \). So, \( 4 \frac{5}{6} + 2 \frac{1}{8} \approx 7 \).
\( 4 \frac{5}{6} \approx 5 \), \( 2 \frac{1}{8} \approx 2 \). So, \( 4 \frac{5}{6} + 2 \frac{1}{8} \approx 7 \).
9. Multiplying Fractions and Whole Numbers
\( 3 \times \frac{2}{5} = \frac{6}{5} = 1 \frac{1}{5} \)
\( 3 \times \frac{2}{5} = \frac{6}{5} = 1 \frac{1}{5} \)
10. Multiplying Fractions: Reciprocals
The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
\( \frac{2}{3} \times \frac{3}{2} = 1 \)
The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
\( \frac{2}{3} \times \frac{3}{2} = 1 \)
11. Multiplying Fractions and Mixed Numbers: Reducing
\( 2 \frac{1}{2} \times \frac{4}{5} = \frac{5}{2} \times \frac{4}{5} = \frac{20}{10} = 2 \)
\( 2 \frac{1}{2} \times \frac{4}{5} = \frac{5}{2} \times \frac{4}{5} = \frac{20}{10} = 2 \)
12. Dividing Fractions by Whole Numbers
\( \frac{3}{4} \div 2 = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} \)
\( \frac{3}{4} \div 2 = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} \)
13. Dividing Whole Numbers by Fractions
\( 4 \div \frac{2}{3} = 4 \times \frac{3}{2} = 6 \)
\( 4 \div \frac{2}{3} = 4 \times \frac{3}{2} = 6 \)
14. Dividing Fractions by Fractions
\( \frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} = 1 \frac{1}{4} \)
\( \frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} = 1 \frac{1}{4} \)
15. Dividing Mixed Numbers
\( 2 \frac{1}{2} \div 1 \frac{1}{4} = \frac{5}{2} \div \frac{5}{4} = \frac{5}{2} \times \frac{4}{5} = 2 \)
\( 2 \frac{1}{2} \div 1 \frac{1}{4} = \frac{5}{2} \div \frac{5}{4} = \frac{5}{2} \times \frac{4}{5} = 2 \)
16. Ratios
12 boys and 8 girls: Ratio = 12:8 = 3:2
12 boys and 8 girls: Ratio = 12:8 = 3:2
17. Proportions and Cross-Multiplying
\( \frac{2}{3} = \frac{4}{6} \): \( 2 \times 6 = 3 \times 4 \), both equal 12.
\( \frac{2}{3} = \frac{4}{6} \): \( 2 \times 6 = 3 \times 4 \), both equal 12.
18. Ratio Tables
If 2 pencils cost \$4, how much do 4 pencils cost?
| Pencils | 2 | 4 |
|---------|---|---|
| Cost | 4 | 8 |
Double pencils, double the cost.
If 2 pencils cost \$4, how much do 4 pencils cost?
| Pencils | 2 | 4 |
|---------|---|---|
| Cost | 4 | 8 |
Double pencils, double the cost.
19. Rates
60 km in 2 hours = 30 km/hour
60 km in 2 hours = 30 km/hour
20. Problem-Solving with Proportions
If 3 shirts cost \$18, how much do 5 shirts cost?
Set up: \( \frac{3}{18} = \frac{5}{x} \)
Cross-multiply: \( 3x = 90 \)
\( x = 30 \). So, 5 shirts cost \$30.
If 3 shirts cost \$18, how much do 5 shirts cost?
Set up: \( \frac{3}{18} = \frac{5}{x} \)
Cross-multiply: \( 3x = 90 \)
\( x = 30 \). So, 5 shirts cost \$30.
Chapter 3: Pre-Test (40 Questions)
- Write the ratio of 8 dogs to 12 cats in simplest form.
- Change \( \frac{13}{5} \) to a mixed number.
- Convert \( 3 \frac{2}{7} \) to an improper fraction.
- Add: \( \frac{4}{9} + \frac{2}{9} \)
- Subtract: \( \frac{7}{10} - \frac{3}{10} \)
- Add: \( \frac{1}{4} + \frac{1}{3} \)
- Subtract: \( \frac{5}{6} - \frac{1}{4} \)
- Add: \( 1 \frac{1}{2} + 2 \frac{1}{3} \)
- Subtract: \( 3 \frac{3}{4} - 1 \frac{2}{5} \)
- Estimate the sum: \( 2 \frac{7}{8} + 3 \frac{1}{10} \)
- Multiply: \( 5 \times \frac{3}{8} \)
- What is the reciprocal of \( \frac{4}{7} \)?
- Multiply: \( 1 \frac{1}{3} \times \frac{3}{4} \)
- Divide: \( \frac{5}{8} \div 2 \)
- Divide: \( 7 \div \frac{1}{2} \)
- Divide: \( \frac{3}{5} \div \frac{2}{3} \)
- Divide: \( 2 \frac{1}{4} \div 1 \frac{1}{8} \)
- Simplify the ratio 12:16.
- Solve: \( \frac{4}{5} = \frac{x}{15} \)
- Complete the ratio table: 1:4, 2:___, 3:___, 4:___
- Find the rate: 120 miles in 3 hours.
- Solve: If 6 notebooks cost \$18, what is the cost of 9 notebooks?
- Write \( \frac{14}{3} \) as a mixed number.
- Write \( 2 \frac{1}{5} \) as an improper fraction.
- Add: \( \frac{7}{12} + \frac{1}{3} \)
- Subtract: \( \frac{5}{6} - \frac{1}{3} \)
- Add: \( 2 \frac{3}{4} + 1 \frac{1}{8} \)
- Subtract: \( 4 \frac{2}{5} - 2 \frac{3}{10} \)
- Estimate the difference: \( 6 \frac{7}{9} - 2 \frac{1}{11} \)
- Multiply: \( 6 \times \frac{2}{3} \)
- What is the reciprocal of \( \frac{5}{9} \)?
- Multiply: \( 3 \frac{2}{5} \times \frac{5}{6} \)
- Divide: \( \frac{7}{9} \div 3 \)
- Divide: \( 8 \div \frac{4}{5} \)
- Divide: \( \frac{5}{6} \div \frac{1}{2} \)
- Divide: \( 4 \frac{2}{3} \div 1 \frac{1}{3} \)
- Simplify the ratio 15:20.
- Solve: \( \frac{3}{4} = \frac{y}{12} \)
- Fill in the ratio table: 2:5, 4:___, 6:___, 8:___
- Find the rate: 250 km in 5 hours.
Chapter 4: Questions & Answers
Q: What is a ratio?
A: A ratio compares two or more numbers, showing how many times one value contains or is contained within the other.
A: A ratio compares two or more numbers, showing how many times one value contains or is contained within the other.
Q: How do you add fractions with like denominators?
A: Add the numerators and keep the denominator the same.
A: Add the numerators and keep the denominator the same.
Q: What do you do before adding fractions with unlike denominators?
A: Find a common denominator, convert the fractions, then add.
A: Find a common denominator, convert the fractions, then add.
Q: How do you change an improper fraction to a mixed number?
A: Divide the numerator by the denominator. The quotient is the whole number, and the remainder over the denominator is the fraction part.
A: Divide the numerator by the denominator. The quotient is the whole number, and the remainder over the denominator is the fraction part.
Q: What is a reciprocal?
A: The reciprocal of a fraction is what you multiply it by to get 1. For \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).
A: The reciprocal of a fraction is what you multiply it by to get 1. For \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).
Q: How do you divide fractions?
A: Multiply by the reciprocal of the divisor.
A: Multiply by the reciprocal of the divisor.
Q: How do you solve a proportion?
A: Use cross-multiplication: \( \frac{a}{b} = \frac{c}{d} \) means \( a \times d = b \times c \).
A: Use cross-multiplication: \( \frac{a}{b} = \frac{c}{d} \) means \( a \times d = b \times c \).
Q: What is a rate?
A: A rate compares two quantities with different units, such as miles per hour.
A: A rate compares two quantities with different units, such as miles per hour.
Q: What is a ratio table?
A: A table that shows pairs of numbers that form equivalent ratios.
A: A table that shows pairs of numbers that form equivalent ratios.
Q: How do you estimate sums or differences of mixed numbers?
A: Round each mixed number to the nearest whole number, then add or subtract.
A: Round each mixed number to the nearest whole number, then add or subtract.
Chapter 5: Post-Test (40 Questions)
- Write the ratio of 15 apples to 20 oranges in simplest form.
- Change \( \frac{17}{6} \) to a mixed number.
- Convert \( 4 \frac{1}{5} \) to an improper fraction.
- Add: \( \frac{5}{8} + \frac{1}{8} \)
- Subtract: \( \frac{9}{12} - \frac{5}{12} \)
- Add: \( \frac{2}{5} + \frac{1}{2} \)
- Subtract: \( \frac{7}{9} - \frac{2}{3} \)
- Add: \( 2 \frac{1}{3} + 3 \frac{1}{4} \)
- Subtract: \( 5 \frac{1}{2} - 2 \frac{2}{3} \)
- Estimate the sum: \( 7 \frac{3}{5} + 1 \frac{7}{8} \)
- Multiply: \( 4 \times \frac{2}{7} \)
- What is the reciprocal of \( \frac{6}{11} \)?
- Multiply: \( 2 \frac{1}{4} \times \frac{4}{5} \)
- Divide: \( \frac{3}{7} \div 2 \)
- Divide: \( 9 \div \frac{3}{4} \)
- Divide: \( \frac{2}{3} \div \frac{5}{6} \)
- Divide: \( 3 \frac{1}{2} \div 1 \frac{3}{4} \)
- Simplify the ratio 18:24.
- Solve: \( \frac{5}{8} = \frac{y}{24} \)
- Complete the ratio table: 1:3, 2:___, 3:___, 4:___
- Find the rate: 200 km in 4 hours.
- Solve: If 8 pens cost \$24, what is the cost of 12 pens?
- Write \( \frac{16}{5} \) as a mixed number.
- Write \( 3 \frac{2}{3} \) as an improper fraction.
- Add: \( \frac{3}{10} + \frac{2}{5} \)
- Subtract: \( \frac{7}{8} - \frac{1}{4} \)
- Add: \( 5 \frac{1}{6} + 2 \frac{5}{12} \)
- Subtract: \( 7 \frac{2}{3} - 4 \frac{5}{9} \)
- Estimate the difference: \( 9 \frac{2}{5} - 3 \frac{7}{8} \)
- Multiply: \( 7 \times \frac{3}{4} \)
- What is the reciprocal of \( \frac{8}{13} \)?
- Multiply: \( 2 \frac{3}{8} \times \frac{4}{7} \)
- Divide: \( \frac{9}{10} \div 5 \)
- Divide: \( 10 \div \frac{2}{3} \)
- Divide: \( \frac{7}{8} \div \frac{3}{5} \)
- Divide: \( 6 \frac{1}{2} \div 2 \frac{1}{4} \)
- Simplify the ratio 21:28.
- Solve: \( \frac{2}{5} = \frac{x}{25} \)
- Fill in the ratio table: 3:7, 6:___, 9:___, 12:___
- Find the rate: 360 miles in 6 hours.