Problem 24 - 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.) 2 weeks to 7 days

Step-by-Step Solution

Answer

The simplified ratio that compares 2 weeks to 7 days is 2:1.

Step 1: Convert weeks into days

As there are 7 days in a week, to convert weeks into days an operation of multiplication must be done: \(2 weeks \times 7 (days/week) = 14 days\).

Step 2: Formulate the ratio

Now, the two quantities to compare are both expressed in the same unit of measurement, the days. Therefore, the ratio can be formulated: \(14 days:7 days\).

Step 3: Simplify the ratio

To simplify this ratio, divide both sides of the ratio by 7: \(\frac{14 days}{7}: \frac{7 days}{7}\), resulting in a simplified ratio of \(2:1\). This means for every 2 units (days) in the first quantity, there is 1 unit (day) in the second quantity.

Key Concepts

Understanding Unit Conversion

Unit conversion is the process of changing the units of a measurement without altering its actual value. It allows us to compare quantities using the same units, making ratios and other calculations possible. For example, when you have a quantity in weeks and another in days, converting weeks to days ensures that you can directly compare the two quantities.

In this exercise, we start with two weeks. We know from our understanding of time that one week equals seven days. To convert weeks into days, we multiply the number of weeks by seven.

• Multiply the quantity by the conversion factor (e.g., 7 days per week).
• Apply the multiplication: For 2 weeks, multiply by 7 to get 14 days.

By performing these steps, we transfer the quantity into a form which can be easily compared to another measurement in the same unit.

Simplifying Ratios

Once both quantities are in the same unit (such as days in this example), it's time to formulate and simplify the ratio. A ratio is a way to compare two quantities by division. It tells us how much one quantity is in relation to another.

After converting weeks to days, we obtained the ratio 14 days to 7 days. To simplify, we need to find the greatest common divisor of the two numbers and divide both sides by this number. Simplifying a ratio involves:

• Identifying the greatest common factor (GCF) of the two quantities.
• Dividing both numbers by the GCF. Here, both 14 and 7 are divisible by 7.

This process reduces our original ratio of 14:7 down to 2:1, meaning for every 2 parts from the first quantity, there is 1 part from the second.

Week to Day Conversion

Understanding how to convert weeks to days is essential when comparing time periods measured differently. This basic conversion is frequently used in various applications, from planning events to solving mathematical exercises like this one.

The conversion factor is straightforward: 1 week is equivalent to 7 days. When translating weeks into days:

• Multiply the number of weeks by 7 (days per week).
• This gives you the total number of days.

For instance, 2 weeks becomes 2 times 7, resulting in 14 days. This transformation allows comparisons between various quantities on a similar scale, ensuring clear analysis and discussion.