Question

People often use mixed numbers to compare two quantities or to describe how much something has changed or grown. a. Dion's height is about \(1 \frac{1}{2}\) times his younger brother Jamil's height. Jamil is about 40 inches tall. How tall is Dion? b. Bobbi spends 40 minutes each night practicing her violin. She said, "That's \(1 \frac{1}{3}\) times the amount of time I spent last year." How much time did Bobbi practice each night last year? c. The 1998 population of Seattle, Washington, was about \(6 \frac{3}{4}\) times the 1900 population. Seattle's 1900 population was about 80,000 Estimate Seattle's population in 1998

Short answer

Dion is 60 inches tall. Bobbi practiced 30 minutes each night last year. Seattle's population in 1998 was approximately 540,000.

Step-by-step solution

Understanding the Problem

Read each question carefully to understand what needs to be calculated. We will determine different values based on the given mixed numbers compared to the given basic quantities.

Step 1a: Convert Mixed Number to Improper Fraction

For Dion's height, first convert the mixed number \(1 \frac{1}{2}\) to an improper fraction. \(1 \frac{1}{2} = \frac{3}{2} \).

Step 2a: Multiplication

Multiply Jamil's height (40 inches) by the improper fraction. \( 40 \times \frac{3}{2} = 40 \times 1.5 = 60 \). Dion is 60 inches tall.

Step 1b: Convert Mixed Number to Improper Fraction

For Bobbi's practice time, convert \(1 \frac{1}{3}\) to an improper fraction. \(1 \frac{1}{3} = \frac{4}{3} \).

Step 2b: Division

Divide the current practice time (40 minutes) by the improper fraction to find out last year's practice time. \( 40 \div \frac{4}{3} = 40 \times \frac{3}{4} = 30 \). Bobbi practiced 30 minutes each night last year.

Step 1c: Convert Mixed Number to Improper Fraction

For the Seattle population, convert \(6 \frac{3}{4}\) to an improper fraction. \(6 \frac{3}{4} = \frac{27}{4} \).

Step 2c: Multiplication

Multiply the 1900 population (80,000) by the improper fraction to find the 1998 population. \( 80,000 \times \frac{27}{4} = 80,000 \times 6.75 = 540,000 \). Seattle's population in 1998 was approximately 540,000.

Key concepts

Conversion of Mixed Numbers to Improper Fractions

Mixed numbers combine whole numbers and fractions. For example, the mixed number \(1 \frac{1}{2}\) means 1 whole and \(\frac{1}{2}\). But to use them in calculations, it's often easier to convert mixed numbers to improper fractions. An improper fraction has a numerator larger than its denominator.

Here's the step-by-step conversion:

• Multiply the whole number by the denominator of the fraction part.
• Add the numerator to this product.
• Your result becomes the numerator of the new fraction, with the original denominator remaining the same.

Let's convert \(1 \frac{1}{2}\) to an improper fraction:

• Multiply 1 (the whole number) by 2 (the denominator): 1 × 2 = 2
• Add 1 (the numerator): 2 + 1 = 3
• Your improper fraction is \(\frac{3}{2}\)

Apply these steps to other mixed numbers similarly, and you'll get the improper fraction needed for further calculations.

Multiplication with Fractions

Multiplying fractions, including improper fractions, follows a simple procedure. It's essential for solving many real-life problems, such as determining proportions or comparing quantities.

Here's the process to multiply fractions:

• Convert any mixed numbers to improper fractions as explained earlier.
• Multiply the numerators (top numbers) together to get the new numerator.
• Multiply the denominators (bottom numbers) together to get the new denominator.

Let's take Dion's height as an example:

• Convert \(1 \frac{1}{2}\) to \(\frac{3}{2}\).
• Multiply Jamil's height (40 inches) by \(\frac{3}{2}\).
• Work through the multiplication: \(40 \times \frac{3}{2} = \frac{40 \times 3}{2} = \frac{120}{2} = 60\)

Therefore, Dion is 60 inches tall. Keep practicing these steps with different problems to get comfortable with multiplication involving fractions.

Division with Fractions

Division with fractions can be straightforward once you understand the rules. It's often used in everyday scenarios, such as splitting quantities or calculating rates.

To divide by a fraction, you need to multiply by its reciprocal. The reciprocal of a fraction is created by swapping its numerator and denominator.

• Convert any mixed numbers to improper fractions first.
• Find the reciprocal of the divisor (the fraction you're dividing by).
• Multiply the dividend by the reciprocal of the divisor.

For example, for Bobbi's practice time:

• She currently practices 40 minutes, which is \(1 \frac{1}{3}\) times what she did last year.
• Convert \(1 \frac{1}{3}\) to \(\frac{4}{3}\).
• Reciprocal of \(\frac{4}{3}\) is \(\frac{3}{4}\).
• Multiply 40 by \(\frac{3}{4}\): \(40 \times \frac{3}{4} = \frac{40 \times 3}{4} = \frac{120}{4} = 30\).

So, Bobbi practiced 30 minutes each night last year. This method simplifies division involving fractions, making it manageable and less intimidating.

Real-Life Math Applications

Understanding conversions, multiplications, and divisions involving fractions is crucial for solving everyday problems. Here are some relatable examples:

• **Cooking:** Recipes often require mixed numbers, such as \(1 \frac{1}{2}\) cups of an ingredient. Accurate conversion to improper fractions ensures the correct proportion of ingredients.
• **Construction:** Measurements often involve mixed numbers. Converting these to improper fractions simplifies precise calculations for materials needed.
• **Budgeting:** Time and money management can involve fractions. For instance, allocating \(\frac{3}{4}\) of your budget to essentials ensures you don't overspend.
• **Healthcare:** Dosages often require precise calculations. For example, a medication amount of \(2 \frac{1}{2}\) might need conversion for accurate administration.

Consider the Seattle population example:

• A historical population of 80,000 in 1900 grew about \(6 \frac{3}{4}\) times by 1998.
• Convert \(6 \frac{3}{4}\) to \(\frac{27}{4}\).
• Multiply: \(80,000 \times \frac{27}{4} = 80,000 \times 6.75 = 540,000\).

Seattle's 1998 population was approximately 540,000. Real-life math applications demonstrate the practicality of mastering fractions.