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In a certain retirement program, benefits are based on retiring at age 62, but if retirement is delayed, the monthly payment is increased by \(\frac{2}{3}\) of \(1\%\) for each month worked after age 62 until age 65. If Mr. Johnson would receive a monthly payment of $310 by starting benefits on his 65th birthday, what would the monthly payment be if he started on his 62nd birthday?
Let's break down what's happening in plain English. Mr. Johnson has a retirement benefit that works like this: he gets a base monthly payment if he retires at age 62. But if he waits longer, his monthly payment gets bigger - specifically, it increases by 2/3 of 1 percent for every month he delays retirement.
Think of it like a savings account that grows each month he waits. We know that by age 65, his monthly payment would be $310. We need to figure out what his base payment would have been at age 62 - before all those monthly increases were added.
Process Skill: TRANSLATE - Converting the retirement benefit language into a clear mathematical relationship
First, let's figure out how many months Mr. Johnson would wait between age 62 and age 65.
From age 62 to age 65 = 3 years
3 years × 12 months per year = 36 months
Now, for each of these 36 months, his payment increases by 2/3 of 1 percent.
\(\frac{2}{3} \times 1\% = 0.667\%\)
Total percentage increase over 36 months:
\(36 \text{ months} \times 0.667\% \text{ per month} = 24\%\)
So by age 65, his monthly payment has increased by 24% compared to what it would have been at age 62.
Here's the key insight: the $310 he receives at age 65 represents his original base payment PLUS the 24% increase we just calculated.
In other words: Base Payment + 24% of Base Payment = $310
We can think of this as: \(\mathrm{Base\,Payment} \times (1 + 24\%) = \$310\)
Or: \(\mathrm{Base\,Payment} \times 1.24 = \$310\)
This is like knowing that some number, when increased by 24%, gives us $310, and we need to find that original number.
Process Skill: INFER - Recognizing that we need to work backwards from the final amount to find the original
Now we can solve for the base payment:
\(\mathrm{Base\,Payment} \times 1.24 = \$310\)
\(\mathrm{Base\,Payment} = \$310 \div 1.24\)
\(\mathrm{Base\,Payment} = \$250\)
Let's verify this makes sense: If Mr. Johnson starts with $250 at age 62, and this increases by 24% over 36 months:
\(\$250 \times 1.24 = \$310\) ✓
This confirms our answer.
The monthly payment if Mr. Johnson started benefits on his 62nd birthday would be $250.00.
Looking at our answer choices, this matches choice B: $250.00.
1. Misunderstanding the direction of the relationship
Many students incorrectly assume that $310 is the base payment at age 62, and they need to calculate what the increased payment would be at age 65. This fundamental misreading of the problem leads them down the wrong path entirely. The question clearly states that $310 is what he would receive by starting benefits on his 65th birthday, but students often miss this critical detail.
2. Confusion about what "2/3 of 1 percent" means
Students frequently struggle with interpreting "2/3 of 1 percent" and may calculate it incorrectly as 2/3 percent (0.67%) instead of (2/3) × (1%) = 0.667%. Some students might even interpret this as 2/3 divided by 1%, leading to completely incorrect percentage calculations.
3. Misunderstanding when the increases are applied
Some students incorrectly think that the 2/3 of 1% increase is applied only once at the end, rather than understanding that it's a monthly compounding increase. However, the solution treats this as a simple cumulative increase (36 months × 0.667% = 24%), which is the correct interpretation for this problem context.
1. Incorrect time period calculation
Students may miscalculate the number of months between age 62 and 65. Some might calculate only 2 years (24 months) instead of 3 years (36 months), or they might get confused about whether to include the starting or ending month in their calculation.
2. Arithmetic errors in percentage calculations
When calculating 2/3 of 1%, students often make computational mistakes. They might calculate 2/3 × 1% as 0.33% instead of 0.667%, or when multiplying by 36 months, they might arrive at 12% or 20% instead of the correct 24%.
3. Errors in the reverse calculation
Once students correctly identify that they need to work backwards from $310, they often make mistakes in the division. They might incorrectly calculate $310 ÷ 1.24, or they might set up the equation wrong by using $310 × 0.24 instead of $310 ÷ 1.24.
No likely faltering points - once students correctly calculate $250, it directly matches answer choice B without any additional interpretation needed.