The Other Three Parts
On the previous page, you looked at a pizza where all that was left was 1/4 or 25%.
What part of this pizza has been eaten? Can you figure out a fraction and a percent to describe how much has been eaten?
Because the picture shows four parts that are all the same size, each part is 1/4 or 25%.
There are three pieces missing, so the amount that is missing can be thought of as
1/4 + 1/4 + 1/4
or
25% + 25% + 25%
When you have three pieces that are each 1/4, you can write that as the fraction 3/4 (three-fourths or three quarters).
When you add 25% + 25% + 25%, that comes to 75%.
The amount of the pizza that has already been eaten is 3/4 or 75%.
Just as you explored different ways of writing fractions that are equal to 1/2 and 1/4, you can also write other fractions that are equal to 3/4. Try using the model below to find a pattern in fractions that are equal to 3/4. Remember that the numbers in the boxes tell you the size of the boxes and you can also look at the number line to see the sizes of the part and the whole. (On this model, you have to pull the orange dot a little farther to get it to move.)
Here are some more fractions that are equal to 3/4. Do they fit your pattern?
Try to write some other fractions that are equal to 3/4 or 75%. How can you tell a fraction is equal to 75%?
(Click here for some explanation and some more examples of fractions that are equal to 75%.)
(Click here for some explanation and some more examples of fractions that are equal to 75%.)
More About 75%
On the previous web page, you explored the idea of 25%. Before further exploring 75%, here are some important ideas about 25% that are worth reviewing:
Now let's look at what 75% looks like. Use the model below to explore. In this model, you can drag the orange dot to change 75% and drag the black dot to change 100% (the whole). As you play with the model, think about how 75% is related to the other percents you have explored so far (50%, 25%, 100%).
- 25% of a number is the same as 1/4 of that number.
- You can find 25% of a number by dividing it by 4.
- You can find 25% of a number by finding half (50%) of it and then finding half (50%) of the result.
- 25% of 100 is 25.
- 25% of $1.00 is 25 cents or $0.25.
- If you take away 25% of something, what is left will be 75% because the whole thing is 100%.
Now let's look at what 75% looks like. Use the model below to explore. In this model, you can drag the orange dot to change 75% and drag the black dot to change 100% (the whole). As you play with the model, think about how 75% is related to the other percents you have explored so far (50%, 25%, 100%).
Use the model to answer these questions:
1. What is 75% of 120?
2. What is the whole if 45 is 75%?
3. If 75% of a number is 90, what is 25% of that same number?
4. If 75% of a number is more than 80, what could the number be?
Click here to check your answers.
1. What is 75% of 120?
2. What is the whole if 45 is 75%?
3. If 75% of a number is 90, what is 25% of that same number?
4. If 75% of a number is more than 80, what could the number be?
Click here to check your answers.
A more flexible model:
The model below works a little differently from the ones you've played with so far. In this one, you can move the orange dot to shade different percents! Play around with both dots to see how it works. Can you tell what each number is telling you? Read below the model if you aren't sure.
The model below works a little differently from the ones you've played with so far. In this one, you can move the orange dot to shade different percents! Play around with both dots to see how it works. Can you tell what each number is telling you? Read below the model if you aren't sure.
(In this model, the red and black numbers below the number line tell you the values of the part and the whole on the number line. The blue percent above the number line tells you what percent the orange part is of the whole rectangle.)
5. Use the model to fill in the first four rows of this table (the yellow ones).
6. What can you say about the relationships between 25%, 50%, 75%, and 100% of a number?
7. Try to use what you learned about the relationships to fill in the last four rows (the green ones). (Note: you can fill in the boxes in the order that makes the most sense to you. You can check your answers with the model!)
Click here to see the filled in table and some things you might have noticed.
Here are some challenges to try with the model above:
6. If 180 people were surveyed and 135 of them said they watch more than 3 hours of tv a day, what percent of people said they watch more than 3 hours of tv a day?
7. Imagine you want to buy something that costs $140 and you have a coupon for 25% off. Can you use the model to figure out how much you will pay? (Hint: Try making the whole $140 and moving the orange dot to 100%, then go back by 25%.)
8. What if the store is having a clearance sale and the thing that cost $140 is 75% off (but you can't use your coupon)? How much will it cost then?
9. Imagine you own a store and you want to raise the prices by 25%. If something costs $60 now, how much will it cost after you raise the price? (Hint: You can move the orange dot past 100%. Look all the way down at the bottom of the page if you want another hint.)
10. If the price of something is $150 after it has been increased by 25%, what was the price before it was increased? (Hint: Try to find a number that will make the orange dot land on 150 after you add 25%. Look all the way down at the bottom of the page if you want another hint.)
11. Which is more: 25% of 180 or 75% of 120? Try to guess first and then use the model to see if you were right.
Estimation challenges:
Sometimes the numbers involved in thinking about percents are messy and you may not need an exact answer. Estimate the answers to these percent questions. To estimate, try changing the numbers to numbers that are close to the given numbers but easier to work with. Make your estimates and then use the model to see how close you got.
12. About how much is 25% of 143?
13. About how much is 75% of 175?
14. If 14 out of 20 people in a room are wearing glasses, what percent is that close to? Is it more or less than that percent?
15. If 23% of a number is 15, about how big is the number?
Click here for answers and explanations for numbers 6-15.
Percents on Circle Graphs - Matching
At this point, you have explored several different percents. When you get to know a percent very well, you can use it as a benchmark. A benchmark is something you know well that you can compare other things to. To review what your benchmark percents look like as circle graphs, try this matching exercise. For each of the percents below, identify the circle graph that shows that the given percent of people prefer paper. (The answers are at the bottom of the page.)
Something to think about: Look at the sliders underneath the graphs. Can you see a connection between them and the model at the top of the page?
A Survey About Reading - Graphing the Results
Now you're ready to make some graphs of the results of the survey. For each town, drag the slider on the circle graph below to show what the survey results for that town look like. Don't worry about getting it exactly right. You're just practicing with benchmark percents. You can use the circle graphs above for reference and also the models at the top of the page. If you want to check your answers, make a sketch (quick drawing) of each of your graphs and then click below the graph to get the answers.)
Survey Results | |||
---|---|---|---|
Town | People surveyed | People who prefer paper | People who prefer a device |
Springfield | 100 | 40 | 60 |
Chester | 120 | 95 | 25 |
Greenville | 65 | 15 | 50 |
Arlington | 75 | 40 | 35 |
(Click here to see some example answers.)
Some hints and answers:
Second hint for question 9: Each of the four blocks inside the whole is worth 25%. To add on 25%, move the orange dot so you make one more block that is the same size.
Second hint for question 10: Every block that is worth 25% is the same size. Move the orange dot to 150 and then move the black dot so that the orange piece outside the whole is the same size as the four pieces inside. Check to see if your picture shows starting with your new whole and adding on 25% like you did in question 8.
Answers to matching exercise: 1-C, 2-A, 3-D, 4-B, 5-E
Second hint for question 9: Each of the four blocks inside the whole is worth 25%. To add on 25%, move the orange dot so you make one more block that is the same size.
Second hint for question 10: Every block that is worth 25% is the same size. Move the orange dot to 150 and then move the black dot so that the orange piece outside the whole is the same size as the four pieces inside. Check to see if your picture shows starting with your new whole and adding on 25% like you did in question 8.
Answers to matching exercise: 1-C, 2-A, 3-D, 4-B, 5-E