Question 6·Hard·Percentages
According to the 2010 Census, the adult population aged 18 years or greater of the United States in 2010 was 234,564,071. In 2010, a survey was conducted among a randomly chosen sample of adults aged 18 years or greater in the United States about their preference to live in a warm climate or a cool climate. The table below summarizes the survey results.
| Climate Preferences | Warm | Cool | No preference | Total |
|---|---|---|---|---|
| 18–35 years old | 295 | 168 | 45 | 508 |
| 36–50 years old | 246 | 123 | 41 | 410 |
| 51–65 years old | 238 | 117 | 48 | 403 |
| Greater than 65 years old | 137 | 78 | 64 | 279 |
| Total | 916 | 486 | 198 | 1,600 |
Which of the following is closest to the difference between the percentage of adults aged 18–50 years who responded "warm" and the percentage of adults aged 51 years or greater who responded "warm"?
For table-and-percentage questions, first translate the verbal description into specific rows you need from the table, then combine rows if a range of ages or categories is given. Next, for each group, compute the relevant percentage by dividing the count of interest by the total in that group. Finally, compare or subtract those percentages and match the result to the closest answer choice, estimating reasonably to save time rather than doing long exact division when approximate values are enough.
Hints
Identify which rows to combine
You are comparing ages 18–50 versus 51 and older. Which two rows in the table together make 18–50, and which two make 51+?
Add the warm responses and totals for each group
For each combined age group (18–50 and 51+), add the "Warm" counts together and the "Total" counts together. You should end up with one warm count and one total count for each big group.
Turn fractions into percentages, then compare
Convert each group's warm fraction (warm ÷ total) into a decimal or percent. Then subtract the smaller percent from the larger percent and see which answer choice is closest to that difference.
Desmos Guide
Use Desmos to compute the percentage difference
In Desmos, type the expression 541/918 - 375/682. The output is the difference between the two warm fractions as a decimal. Multiply that decimal by 100 in your head (or type (541/918 - 375/682)*100) to see the difference in percentage points and then compare that value to the answer choices.
Step-by-step Explanation
Group the correct age ranges and add counts
The question compares adults aged 18–50 years with adults aged 51 years or greater.
From the table:
- Ages 18–35 and 36–50 together make the 18–50 group.
- Warm:
- Total:
- Ages 51–65 and greater than 65 together make the 51+ group.
- Warm:
- Total:
Compute the warm percentage for each age group
For each age group, divide the number who said "warm" by the total in that group.
-
Ages 18–50:
- Fraction warm:
- This is approximately , or about .
-
Ages 51 and older:
- Fraction warm:
- This is approximately , or about .
Find and match the difference to an answer choice
Now subtract the percentages:
- Difference .
So the difference between the percentage of 18–50-year-olds who responded "warm" and the percentage of adults 51 or older who responded "warm" is closest to 4%, which matches answer choice C) 4%.