Problem 62 - Elementary Algebra Within Reach

Question

Find a ratio that compares the relative sizes of the quantities. (Use the same units of measurement for both quantities.) \(5 \frac{1}{2}\) pints to 2 quarts

Step-by-Step Solution

Answer

The ratio comparing the two quantities is \(11:8\).

Step 1: Convert All Quantities to the Same Unit

Start by aligning the units. Convert 5 1/2 pints to quarts since it's easier to convert pints to quarts. Knowing that 1 quart equals 2 pints, divide 5 1/2 (which is 5.5 in decimal form) by 2: \(5.5 \div 2 = 2.75\) quarts.

Step 2: Construct a Ratio

Now that both quantities are in the same unit (quarts), construct a ratio. That is: \(2.75 : 2\), which simplifies to \(11 : 8\) by multiplying both numbers by 4 to remove the decimal from 2.75.

Step 3: Express the Ratio in Simplest Form

Express the ratio in its simplest form, which is already \(11:8\).

Key Concepts

Unit Conversion

Unit conversion is essential when comparing different quantities because it ensures that we are comparing similar things. Think of it as converting apples to apples before seeing how many you have relative to oranges.
This process involves changing the units of measurement of a given quantity to another unit while keeping the value intact. In the exercise, we needed to convert pints to quarts to have a fair comparison.

• First, understand the relationship between units: in this example, 1 quart equals 2 pints.
• Then, apply the conversion factor: convert pints to quarts by dividing the pint amount by 2, since 2 pints make a quart.

Converting ensures that you're not comparing, for example, apples to oranges, but instead, both quantities in the same terms.

Simplifying Ratios

Simplifying ratios is like streamlining information so it's easy to understand at a glance. It's about finding a clean, straightforward expression to convey how much of one thing we have compared to another.

• Take the ratio you obtained after unit conversion. In our example, we first found the ratio was \(2.75 : 2\).
• To simplify, convert any fractions or decimals involved into whole numbers if possible. Here, multiplying both sides of the ratio by 4 removed the decimal, transforming \(2.75 : 2\) into \(11:8\).

Simplifying ensures that your ratios are expressed in the simplest whole number form, making them easier to interpret and compare.

Measurement Comparison

Measurement comparison is the final step in understanding the relationship between two or more quantities. It's the outcome of unit conversion and ratio simplification.
By the end, you should be able to clearly state and understand the proportionate size difference or relationship.

• After simplifying, the ratio, like \(11:8\), tells you how one measurement compares to the other in simplest terms.
• This step is crucial for making informed decisions or evaluations, as it provides a clear and concise understanding of the difference or proportion.

In our particular case, the ratio \(11:8\) tells us there are 11 parts of the first measurement for every 8 of the second. This clear statement allows for easy analysis and simple communication of measurement relations.