Chapter 1 Notes Calculator Resource: Desmos Scientific Calculator • Rounding: Rounding Decimals on the Desmos Scientific Calculator [+] • Fractions: Fraction Operations on the Desmos Scientific Calculator [+] Using the Factor Label Method Your car can drive 370 miles on a tank of 29 gallons. How far can it drive on 49 gallons? Round your answer to the nearest mile. 49 gallons 1 × = How many gallons are needed to drive 1,320 miles? Round your answer to the nearest tenth of a gallon. 1,320 miles 1 × = Video 1 Video 2 Post to forum
The Correct Answer and Explanation is:
Here are the correct answers and a detailed explanation for the problems.
Problem 1: How far can it drive on 49 gallons?
- Filled-in equation:
(49 gallons / 1) × (370 miles / 29 gallons) = 625 miles
Problem 2: How many gallons are needed to drive 1,320 miles?
- Filled-in equation:
(1,320 miles / 1) × (29 gallons / 370 miles) = 103.5 gallons
Explanation
The Factor Label Method, also known as dimensional analysis, is a technique for converting units. The core principle is to multiply a given quantity by one or more conversion factors to arrive at the desired unit. A conversion factor is a fraction that is equal to one because its numerator and denominator represent the same value in different units.
The problem provides the base conversion rate: the car travels 370 miles for every 29 gallons of gas. This can be written as two possible conversion factors:
(370 miles / 29 gallons) or (29 gallons / 370 miles)
Solving the First Problem
The goal is to find the distance in miles a car can travel on 49 gallons. We start with the given quantity, 49 gallons. To convert this to miles, we need to multiply it by a conversion factor that has gallons in the denominator. This will allow the “gallons” unit to cancel out, leaving “miles” as the final unit.
The correct setup is:
(49 gallons / 1) × (370 miles / 29 gallons)
Here, the “gallons” in the numerator of the first term cancels with the “gallons” in the denominator of the second term. The calculation is:
(49 × 370) / 29 = 18,130 / 29 ≈ 625.172 miles
The question asks to round to the nearest mile, so the answer is 625 miles.
Solving the Second Problem
The goal is to find the number of gallons needed to drive 1,320 miles. We start with the given quantity, 1,320 miles. To convert this to gallons, we must use a conversion factor with “miles” in the denominator to cancel the starting unit.
The correct setup is:
(1,320 miles / 1) × (29 gallons / 370 miles)
The “miles” unit cancels out, leaving “gallons” as the final unit. The calculation is:
(1,320 × 29) / 370 = 38,280 / 370 ≈ 103.459 gallons
The question asks to round to the nearest tenth of a gallon, so the answer is 103.5 gallons.
