Question 9·Hard·Two-Variable Data: Models and Scatterplots
Two investments were made as shown in the table below.
| Amount invested | Balance increase | |
|---|---|---|
| Account A | $500 | 6% annual interest |
| Account B | $1,000 | $25 per year |
The interest in Account A is compounded once per year. Which of the following is true about the investments?
For questions comparing investments, first turn any percentages into actual dollar amounts using the given starting balances so you can compare concrete numbers (for example, find 6% of $500). Then note whether each account’s yearly earnings stay constant or change over time—simple interest usually means a fixed dollar amount per year, while compound interest at a fixed percentage means the dollar amount grows as the balance grows. Use one or two specific years (like year 1 and year 2) to see who starts ahead and whether the pattern is increasing or flat, and then match that behavior to the verbal descriptions in the answer choices without doing unnecessary long-term calculations.
Hints
Translate the interest descriptions into yearly earnings
Focus on what "6% annual interest, compounded once per year" means for Account A and what "$25 per year" means for Account B. How does each account’s balance change at the end of a year?
Compare the first year of interest
Calculate how much interest Account A earns in the first year by finding 6% of $500. Then compare that to the $25 Account B earns in a year.
Think about later years
Does Account B’s yearly earning change from year to year? For Account A, since 6% is taken on a growing balance, what happens to the dollar amount of interest each year as the balance gets larger?
Match the behavior to the answer choices
Once you know whether each account’s yearly earnings stay the same or change over time, decide if Account A is ever behind Account B, ahead at first then behind later, behind at first then ahead later, or always ahead.
Desmos Guide
Model each account’s yearly earnings
In Desmos, define the yearly earnings for each account as functions of the year number :
- Type:
A(x) = 30*(1.06)^(x-1)(Account A’s earnings each year, starting at $30 when ) - Type:
B(x) = 25(Account B’s constant yearly earning)
Adjust the viewing window to see several years
Use the wrench (graph settings) to set the -axis from about 1 to 20 and the -axis from about 0 to 60 so you can clearly see both graphs over many years.
Interpret the graphs to choose an answer
Look at the two curves: the horizontal line for and the increasing curve for . Check which function is higher in the first year and whether the two graphs ever intersect or switch positions as increases. Then select the answer choice whose description of "earns more per year" over time matches what you see.
Step-by-step Explanation
Interpret what each account’s description means
From the table:
- Account A: "$500, 6% annual interest, compounded once per year" means that at the end of each year, the bank adds 6% of the current balance to the account. The balance grows, and each year’s interest is 6% of whatever the balance is at the start of that year.
- Account B: "$1,000, $25 per year" means the account’s balance increases by the same fixed dollar amount, $25, every year (simple, not percentage-based growth).
Find how much each account earns in the first year
Compute the money earned in the first year for each account.
- Account A: 6% of $500 is
So Account A earns $30 in the first year.
- Account B: The problem states it earns $25 per year, so it earns $25 in the first year.
So, in the first year, Account A earns more money than Account B.
Understand how the yearly earnings change over time
Now look at how each account’s yearly earnings behave over many years.
- Account B: Always earns exactly $25 every year. The yearly earning is constant.
- Account A: At the end of year 1, the balance is $500 + $30 = $530. In year 2, the interest is 6% of $530:
In year 3, the balance is larger again, so 6% of that new, larger balance will be more than $31.80, and so on.
So the amount Account A earns each year keeps increasing from $30 upward, while Account B stays at $25 every year.
Compare the patterns and match them to the answer choices
You found that:
- In the first year, Account A earns $30 and Account B earns $25, so Account A earns more at the start.
- After that, Account A’s yearly earnings increase from $30 to $31.80 and higher, while Account B always earns exactly $25 per year.
Since Account A starts by earning more and its yearly earnings only go up from there, Account B can never earn more per year than Account A.
Therefore, the true statement is:
Account A always earns more money per year than Account B.