Question

You can use 100 -grids to model real-world problems involving percents. In Parts a and b, show how you could use a 100 -grid to model and solve the problem. a. The sixth grade has raised \(\$ 880\) of the \(\$ 2,000\) they need for a class trip. What percent of the \(\$ 2,000\) do they still need to raise? b. Challenge This year, tickets to the dance cost \(\$ 15 .\) This is \(125 \%\) of last year's cost. How much did tickets cost last year?

Short answer

They still need to raise 56\texttt{\textpercent} of the total amount. Last year's tickets cost \$12.

Step-by-step solution

- Establishing the Part-to-Whole Ratio (Part a)

First, determine the total amount needed and the amount already raised. The formula we use is: \[ \text{Percent Still Needed} = \left( 1 - \frac{\text{Amount Raised}}{\text{Total Amount Needed}} \right) \times 100 \].

- Plug in the Values (Part a)

Substitute \$880 for the amount raised and \$2,000 for the total amount needed:\[ \text{Percent Still Needed} = \left( 1 - \frac{880}{2000} \right) \times 100 \].

- Simplify the Expression (Part a)

First, calculate the fraction:\[ \frac{880}{2000} = 0.44 \]. Then subtract from 1:\[ 1 - 0.44 = 0.56 \]. Finally, multiply by 100 to convert to a percentage:\[ 0.56 \times 100 = 56\text{\texttt{\textpercent}} \]. The sixth grade still needs to raise 56\texttt{\textpercent} of the total amount.

- Understand the Given Information (Part b)

This year, tickets cost \$15, which is 125\texttt{\textpercent} of last year's cost. We need to set up an equation to find last year's cost. The formula we use is: \[ \text{This Year's Cost} = 1.25 \times \text{Last Year's Cost} \].

- Solve for Last Year's Cost (Part b)

Let \( x \) be the cost of tickets last year. Then the equation becomes: \[ 15 = 1.25x \]. To find \( x \), divide both sides by 1.25: \[ x = \frac{15}{1.25} \].

- Calculate the Value (Part b)

Perform the division: \[ \frac{15}{1.25} = 12 \]. Last year's tickets cost \$12.

Key concepts

percent calculations

Percent calculations help us understand proportions and quantities in terms of a hundred. They are used extensively in various real-life scenarios, from finance to academics.
For instance, when calculating the percentage of a goal achieved or remaining, we break it into parts and wholes.
Consider this example:
The sixth grade has raised \$880\ of the \$2000\ they need for a class trip.
Using the formula for the percentage still needed, we have:
\[\text{Percent Still Needed} = \left( 1 - \frac{\text{Amount Raised}}{\text{Total Amount Needed}} \right) \times 100\]
Plugging in the values:
\[ = \left(1 - \frac{880}{2000}\right) \times 100\]
Calculating the fraction:
\[\frac{880}{2000} = 0.44\]
Then:
\[1 - 0.44 = 0.56\]
Finally:
\[0.56 \times 100 = 56\text{\texttt{\textpercent}}\]
So, they still need to raise 56\text{\texttt{\textpercent}} of the total amount.

real-world math applications

Math is crucial in solving everyday problems. Understanding percentages can help in budgeting, shopping, and much more.
For example, if tickets to a dance cost \$15\ this year, which is \125 \text{\texttt{\textpercent}}\ of last year's cost, figuring out last year's price helps in planning expenses.
Using the formula:
\[\text{This Year's Cost} = 1.25 \times \text{Last Year's Cost}\]
We substitute the known value (\$15 for this year's cost) and solve for last year's cost:
\[15 = 1.25 \times \text{Last Year's Cost}\]
Let \(x\) be last year's cost:
\[15 = 1.25x\]
Dividing both sides by 1.25:
\[x = \frac{15}{1.25}\]
Calculating the value:
\[\frac{15}{1.25} = 12\]
Thus, last year's tickets cost \$12.\

part-to-whole ratio

The part-to-whole ratio is key in understanding how a part relates to its whole in percentage calculations.
In a financial goal example, if \$880\ of a \$2000\ goal is raised, finding the percentage of the amount still needed follows:
Using formula:
\[\text{Percent Still Needed} = \left(1 - \frac{880}{2000}\right) \times 100\]
This equation shows the remaining part-to-whole ratio in percent.
First, calculate the fraction:
\[\frac{880}{2000} = 0.44\]
Then:
\[1 - 0.44 = 0.56\]
Finally, convert it to a percentage:
\[0.56 \times 100 = 56\text{\textpercent}\]
Hence, 56\% still needs to be raised.
Understanding part-to-whole ratios helps in making informed decisions and managing expectations in various scenarios.