Problem 21 - Prealgebra

Question

Travel Expenses A family plans to drive their car during their annual vacation. The car can go 350 miles on a tank of gas, which is 18 gallons of gas. The vacation they have planned will cover \(1,785\) miles. How many gallons of gas will that take? Nutrition A 9-ounce serving of pasta contains 159 grams of carbohydrates. How many grams of carbohydrates do 15 ounces of this pasta contain?

Step-by-Step Solution

Answer

91.8 gallons of gas are needed; 15 ounces of pasta contain 265.05 grams of carbohydrates.

Step 1: Calculate the mileage per gallon

To find out how many miles the car can travel on one gallon of gas, we divide the total miles (350) by the total gallons (18): \[ \text{Miles per gallon} = \frac{350}{18} \approx 19.44 \text{ miles per gallon} \]

Step 2: Calculate total gallons needed for 1,785 miles

Now that we know the car travels approximately 19.44 miles per gallon, we can calculate how many gallons are needed to cover 1,785 miles by dividing the total miles by the miles per gallon: \[ \text{Gallons needed} = \frac{1785}{19.44} \approx 91.8 \text{ gallons} \]

Step 3: Calculate carbohydrates per ounce

To find out how many grams of carbohydrates are in one ounce of pasta, divide the grams of carbohydrates in a 9-ounce serving by the number of ounces: \[ \text{Carbohydrates per ounce} = \frac{159}{9} = 17.67 \text{ grams per ounce} \]

Step 4: Calculate carbohydrates in 15 ounces

Now, to find out how many grams of carbohydrates are in 15 ounces of pasta, multiply the carbohydrates per ounce by 15: \[ \text{Carbohydrates in 15 ounces} = 17.67 \times 15 = 265.05 \text{ grams} \]

Key Concepts

Understanding Unit Rate

Unit rate is a fundamental concept in mathematics that helps us understand how many units of one quantity correspond to a single unit of another quantity. It is essentially a ratio expressed as

• Cost per item
• Miles per gallon
• Grams per ounce

To calculate a unit rate, divide the first quantity by the second. For example, in terms of a car's fuel efficiency, if a car travels 350 miles on 18 gallons of gas, the unit rate of miles per gallon is \[ \text{Miles per gallon} = \frac{350}{18} \approx 19.44 \]This means the car can go approximately 19.44 miles on one gallon of fuel. Understanding unit rates helps in making various comparisons and calculations more manageable. Whether you're comparing prices, speeds, or fuel efficiencies, calculating the unit rate provides a clearer picture by leveling the playing field.

Mileage Calculation for Trips

Mileage calculations are vital when planning trips, especially to determine how much fuel is needed for a certain distance. Calculating miles per gallon (mpg) is one typical way to evaluate a vehicle's fuel efficiency. To find out how many gallons of gas are required for a journey, you first need to know the car's mpg, which is calculated like so:\[ \text{Miles per gallon} = \frac{350}{18} \approx 19.44 \]Once the unit rate (mpg) is known, calculating the total gallons needed for a specific trip involves dividing the total trip miles by the mpg: \[ \text{Gallons needed} = \frac{1785}{19.44} \approx 91.8 \]

• Step 1: Find the unit rate (miles per gallon).
• Step 2: Use the unit rate to determine the total gallons required for your trip.

This process ensures you know how much fuel you will need, preventing surprises when you hit the road.

Carbohydrate Calculation in Nutrition

Carbohydrate calculation is crucial for those monitoring diet and nutrition, be it for health reasons or athletic performance. When you know the carbohydrate content of food in a fixed serving size, you can find how much is present per ounce by calculating the unit rate.Let's take pasta as an example. A 9-ounce serving contains 159 grams of carbohydrates, so the carbohydrates per ounce is calculated as:\[ \text{Carbohydrates per ounce} = \frac{159}{9} = 17.67 \]Using this information, you can find out the carbohydrate content for any serving size. For 15 ounces of pasta:\[ \text{Carbohydrates in 15 ounces} = 17.67 \times 15 = 265.05 \]

• Determine carbohydrates in one ounce.
• Adjust for different serving size by multiplying by the desired number of ounces.

Understanding how to calculate in such ways empowers you to tailor nutrition to specific dietary needs with precision.