Fraction Problems

This section demonstrates how to solve practical problems involving fractions.

Example: Rachel needs to determine how many packs of butter to buy for 80 cupcakes, where each cupcake requires 2/25 of a pack.

The solution involves multiplying the fraction by the number of cupcakes and rounding up to the nearest whole pack:

80 × 2/25 = 160/25 = 6.4 packs

Highlight: When dealing with real-world problems, remember to round up to the nearest whole unit when partial units aren't practical.

Comparing Fractions

This page explains how to compare fractions by finding a common denominator.

Example: To determine which fraction is closer to 1, 2/3 or 9/7, convert both to a common denominator of 21:

2/3 = 14/21 9/7 = 27/21

Subtracting from 1 $21/21$:

21/21 - 14/21 = 7/21 21/21 - 27/21 = -6/21

Highlight: The fraction with the smaller difference from 1 (in absolute value) is closer to 1. In this case, 9/7 is closer to 1 than 2/3.

Fractions, Decimals, and Percentages

This page provides a comprehensive table showing the relationships between fractions, decimals, and percentages.

Example:

• 1/2 = 0.5 = 50%
• 1/4 = 0.25 = 25%
• 3/4 = 0.75 = 75%

Highlight: Understanding these equivalences is crucial for converting fractions, decimals, and percentages in various mathematical contexts.

More Fractions, Decimals, and Percentages

This page continues with additional examples of fraction, decimal, and percentage equivalences.

Example:

• 1/10 = 0.1 = 10%
• 1/5 = 0.2 = 20%
• 3/8 = 0.375 = 37.5%

Highlight: Recognizing these patterns helps in quickly converting between different representations of the same value.

Converting Between Fractions, Decimals, and Percentages

This section explains the process of converting between fractions, decimals, and percentages.

Example: To convert 7/20 to a decimal and percentage:

1. Divide 7 by 20: 7 ÷ 20 = 0.35
2. Multiply by 100 for percentage: 0.35 × 100 = 35%

Highlight: When converting from a decimal to a fraction, place the digits after the decimal point over a power of 10 $e.g., 0.61 = 61/100$.

Rounding Numbers

This page introduces the concept of rounding numbers to a specified decimal place.

Example: Rounding 13.72 to one decimal place:

1. Identify the last digit to keep (7)
2. Look at the next digit (2) as the decider
3. Since 2 < 5, the last digit stays the same Result: 13.7

Highlight: The digit to the right of the rounding position is called the "decider" and determines whether to round up or down.

Rounding Numbers (Continued)

This page provides another example of rounding numbers, this time to two decimal places.

Example: Rounding 7.45839 to two decimal places:

1. Identify the last digit to keep (5)
2. Look at the next digit (8) as the decider
3. Since 8 ≥ 5, round up Result: 7.46

Highlight: When the decider is 5 or greater, round up the last digit to be kept.

Rounding Numbers (Special Cases)

This section covers a special case in rounding where rounding up a 9 is necessary.

Example: Rounding 45.698 to two decimal places:

1. Identify the last digit to keep (9)
2. Look at the next digit (8) as the decider
3. Since 8 ≥ 5, round up 9 to 10
4. Replace 9 with 0 and add 1 to the digit on the left Result: 45.70

Highlight: When rounding up a 9, it becomes 0, and you must add 1 to the next digit to the left.

Significant Figures

This page introduces the concept of significant figures in rounding numbers.

Definition: The first significant figure is the first non-zero digit from the left. Subsequent digits, including zeros, are counted as significant figures.

Example: In 0.002309, the significant figures are: 1st: 2 2nd: 3 3rd: 0 4th: 9

Highlight: Understanding significant figures is crucial for maintaining precision in scientific and engineering calculations.

Rounding to Significant Figures

This section demonstrates how to round numbers to a specified number of significant figures.

Example: Rounding 506.07 to two significant figures:

1. Identify the first two significant digits (50)
2. Look at the next digit (6) as the decider
3. Since 6 ≥ 5, round up Result: 510

Highlight: When rounding to significant figures, you may need to add zeros to maintain the correct magnitude of the number.