RSM was founded in 1996. From 1996 to 1997, the number of students in RSM doubled. From 1997 to 1998, the number of students doubled again. By what percent did the number of students increase from 1996 to 1997?
The Correct Answer and Explanation is:
Let’s analyze the problem step-by-step.
Problem restatement:
- RSM was founded in 1996.
- From 1996 to 1997, the number of students doubled.
- From 1997 to 1998, the number of students doubled again.
We are asked:
By what percent did the number of students increase from 1996 to 1997?
Step 1: Understand what “doubled” means.
If the number of students doubles, it means the new number is 2 times the original number.
Step 2: Define variables.
Let the number of students in 1996 be SSS.
From 1996 to 1997, the number of students doubled:
Number of students in 1997 = 2×S=2S2 \times S = 2S2×S=2S.
Step 3: Find the percent increase from 1996 to 1997.
Percent increase formula:Percent Increase=(New Value−Original ValueOriginal Value)×100%\text{Percent Increase} = \left( \frac{\text{New Value} – \text{Original Value}}{\text{Original Value}} \right) \times 100\%Percent Increase=(Original ValueNew Value−Original Value)×100%
Plug in values:Percent Increase=(2S−SS)×100%=(SS)×100%=100%\text{Percent Increase} = \left( \frac{2S – S}{S} \right) \times 100\% = \left( \frac{S}{S} \right) \times 100\% = 100\%Percent Increase=(S2S−S)×100%=(SS)×100%=100%
Step 4: Interpretation.
The number of students increased by 100% from 1996 to 1997.
Additional explanation:
- Doubling means the number becomes twice as large.
- A 100% increase means the quantity has increased by an amount equal to the original amount.
- For example, if there were 100 students in 1996, doubling would mean 200 students in 1997.
The increase is 200−100=100200 – 100 = 100200−100=100 students, which is exactly 100% of 100.
Step 5: Verify consistency with the next year.
From 1997 to 1998, the students again doubled:
Number of students in 1998 = 2×2S=4S2 \times 2S = 4S2×2S=4S.
So overall, from 1996 to 1998, the student body quadrupled (increased by 300%).
Final Answer:
The number of students increased by 100% from 1996 to 1997.
Explanation Summary
The problem states that RSM was founded in 1996, and from 1996 to 1997, the number of students doubled. The phrase “doubled” means that the number of students became twice as many as the previous year. To understand the percentage increase, let’s assign the initial number of students in 1996 as SSS.
Doubling the student number from 1996 to 1997 means the new number of students in 1997 is 2×S=2S2 \times S = 2S2×S=2S. To calculate the percentage increase, we find the difference between the new and original numbers, then divide that difference by the original number, and multiply by 100 to convert it to a percentage. Mathematically, this is:Percent Increase=(2S−SS)×100%=(SS)×100%=100%\text{Percent Increase} = \left(\frac{2S – S}{S}\right) \times 100\% = \left(\frac{S}{S}\right) \times 100\% = 100\%Percent Increase=(S2S−S)×100%=(SS)×100%=100%
Therefore, the student population increased by 100% from 1996 to 1997. This means the increase in the number of students was equal to the original number of students.
The problem also states the number doubled again from 1997 to 1998, meaning the number in 1998 was 2×2S=4S2 \times 2S = 4S2×2S=4S. Thus, the total increase from 1996 to 1998 was from SSS to 4S4S4S, which is a 300% increase over two years.
In conclusion, the percentage increase from 1996 to 1997 is 100%. This reflects a doubling, which is a common expression meaning the value has grown by one full original amount, or increased by 100%.
