Worked Problems
Percentage and Fraction Changes
Tip 1:
Make sure to convert to get equal units when the units used for the two values are not the same.
Problem #1
A container holds 50 liters of water in the morning. In the afternoon, the same container now holds 30 liters of water. By what percentage did the container’s water content decrease?
How I Solve This
I solve this in three steps.
Step 1: Difference.
First, I find the difference between the original water content and the current water content. Since the units are the same, I don’t need to concern myself with them; I’ll just work with the numbers themselves.
50 – 30 = 20
The difference is 20 (or, with units, 20 liters).
Step 2: Ratio.
Next, I find the ratio of the difference to the original water content. This is 20 to 50, or 20/50, or 2/5. This is the same as 40%.
Step 3: Answer.
Finally, I write the answer.
Since the ratio of the difference to the original water content is 40%, the water content decreased by 40%.
Extra Step (For A+ Students): Check.
I can check my answer by finding the ratio of the current water content to the original water content:
30/50 = 3/5 = 60%
So 60% is the amount of water remaining from the original. In other words, the container now holds 60% of the original amount of water. This is the same as saying that the water content reduced by 40%, since 100% – 60% = 40%. Thus, my answer is right.
Problem #2
By what fraction do you need to increase 50 to get 75?
How I Solve This
I solve this very similarly to the previous problem, again in 3 steps.
Step 1: Difference.
The difference between the desired number and original number, which is 75 – 50 = 25.
Step 2: Ratio.
The ratio is that of the difference to the original number (50):
25/50 = 1/2
Step 3: Answer.
This part involves a little more reasoning.
I know that 25 is ½ of 50. I also know that 50 + 25 is 75 (the number I need to get to). Since 25 is ½ of 50, I can rewrite that equation as 50 + ½(50) = 75. Factoring out a 50, I get 50(1 + ½) = 75, or 50(3/2) = 75. This clearly shows me that I’m multiplying 50 by 3/2, or, in other words, that I’m increasing 50 by ½.
Therefore, the answer is that I need to increase 50 by ½ to get 75.
Extra Step (For A+ Students): Check.
To check this problem, I can find the ratio of 75 to 50. It should be equal to 3/2:
75/50 = 3/2
3/2 = 3/2
The answer checks out, so I solved the problem correctly.
Summary of My Method
To solve these kinds of problems, I follow 3 steps:
- Difference: I find the difference between the bigger number and the smaller number. I don’t consider units when the units are the same. Otherwise, I will convert units so they are the same.
- Ratio: I determine the ratio of the difference to the original, starting point.
- Answer: I use the ratio I determined and a little reasoning when needed to write the answer.
Also, I make sure to check my answers to ensure I calculated correctly.
If you have any questions or would like some help with problems like these, feel free to send me a message. I hope these calculations will help you as you continue on your journey of math mastery 🙂
~ Your Humble Study Buddy,
Ace