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Chapter 18: Problem 7

Express the ratios in the simplest form. $$20 \mathrm{qt} \text { to } 2.5 \mathrm{gal}$$

Short Answer

Expert verified
The simplest form of the ratio is 2:1.

Step by step solution

01

Understand the Units

We have two different units in the problem: quarts (qt) and gallons (gal). We need to convert them to the same unit to compare or simplify them.
02

Convert Gallons to Quarts

Since there are 4 quarts in a gallon, convert 2.5 gallons to quarts:\[ 2.5 \text{ gal} \times 4 \text{ qt/gal} = 10 \text{ qt} \]
03

Express the Ratio

Now that both quantities are in quarts, the ratio is 20 qt to 10 qt. We can write this as:\[ \frac{20 \text{ qt}}{10 \text{ qt}} \]
04

Simplify the Ratio

Divide both the numerator and the denominator by their greatest common divisor, which is 10:\[ \frac{20}{10} = 2 \]So, the simplest form of the ratio is 2:1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Conversion
Unit conversion is essential when comparing values that involve different measurement units. In our exercise, we were given quarts and gallons, which require converting to a common unit for comparison. A gallon is a larger unit than a quart. To convert gallons to quarts, remember that there are 4 quarts in every gallon. This is a crucial conversion factor. For example, to convert 2.5 gallons into quarts, you multiply by 4, which gives 10 quarts.
  • Identify the conversion factor (e.g., 1 gallon = 4 quarts).
  • Multiply the number of gallons by the conversion factor to convert to quarts.
Once converted, you can easily compare or simplify ratios. Converting measurement units ensures that each part of a ratio is expressed in the same unit, which is vital for accurate simplification.
Simplification
Simplification is the process of reducing a ratio to its simplest form. It can be done by dividing both terms of the ratio by their greatest common divisor (GCD). When both amounts in the ratio are expressed in the same unit, we can focus on simplifying the numbers themselves. For instance, the ratio 20 quarts to 10 quarts simplifies as follows:
  • Find the GCD of 20 and 10, which is 10.
  • Divide both parts of the ratio by this number.
This results in a simplified ratio of 2:1. Simplification makes ratios easier to interpret and use in calculations, ensuring clarity and precision in mathematical communication.
Measurement Units
Measurement units are standards for expressing physical quantities, and they play a crucial role in calculations. They're necessary for accurately describing amounts of materials, distances, or any quantifiable aspects in our daily lives. In the exercise, we dealt with:
  • Quarts (qt) - a unit of volume.
  • Gallons (gal) - another unit of volume, larger than quarts.
Understanding different measurement units and their conversions is key to solving problems that involve ratios or comparisons. Always keep track of the units you are working with to ensure accurate results. Whether converting, comparing, or simply calculating, clearly understanding and correctly using measurement units will guide you toward correctness in mathematical problem-solving.

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Most popular questions from this chapter

$$\text { Find the required ratios.}$$ Power is defined as the ratio of work done to the time required to do the work. If an engine performs 3.65 kJ of work in 15.0 s, find the power developed by the engine. (See Appendix B.)

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