For the first quarter of 1980, the profits of a certain food-market chain amounted to $49{,}350{,}000. If this figure is...

1. Translate the problem requirements: We know that \(\$49,350,000\) represents a 5% increase over 1979's profits. We need to find the dollar amount of this increase, not the 1979 profits themselves.
2. Set up the percentage relationship: Express the 1980 profits in terms of 1979 profits using the 5% increase relationship.
3. Calculate the 1979 base profits: Work backwards from the 1980 figure to find what the original 1979 profits were.
4. Find the dollar increase: Subtract the 1979 profits from the 1980 profits to get the actual increase amount.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're given and what we need to find in everyday terms.

We know that the food-market chain made \(\$49,350,000\) in the first quarter of 1980. This amount is described as being "5 percent higher than" the profits from the same period in 1979.

Think of it this way: if you got a 5% raise at work, your new salary would be your old salary plus 5% of your old salary. Similarly, the 1980 profits are the 1979 profits plus 5% of the 1979 profits.

The question asks for "the amount of the increase" - this means we need to find how many dollars more the company made in 1980 compared to 1979.

Process Skill: TRANSLATE - Converting the percentage language into a clear mathematical relationship

2. Set up the percentage relationship

Let's express this relationship in plain English first, then write it mathematically.

In words: "1980 profits equals 1979 profits plus 5% of 1979 profits"

We can also think of this as: "1980 profits equals 105% of 1979 profits"

Why 105%? Because 100% represents the original 1979 amount, and we're adding 5% more to get 105% total.

If we call the 1979 profits "P", then:
1980 profits = \(\mathrm{P} + 0.05\mathrm{P} = 1.05\mathrm{P}\)

Since we know 1980 profits = \(\$49,350,000\):
\(\$49,350,000 = 1.05\mathrm{P}\)

3. Calculate the 1979 base profits

Now we need to work backwards to find what P (the 1979 profits) actually was.

We have: \(\$49,350,000 = 1.05\mathrm{P}\)

To find P, we divide both sides by 1.05:
\(\mathrm{P} = \$49,350,000 \div 1.05\)

Let's calculate this step by step:
\(\$49,350,000 \div 1.05 = \$49,350,000 \div (21/20) = \$49,350,000 \times (20/21)\)

\(\$49,350,000 \times (20/21) = \$47,000,000\)

So the 1979 profits were \(\$47,000,000\).

4. Find the dollar increase

Now we can find the actual dollar amount of the increase by subtracting the 1979 profits from the 1980 profits.

Increase = 1980 profits - 1979 profits
Increase = \(\$49,350,000 - \$47,000,000\)
Increase = \(\$2,350,000\)

Let's verify this makes sense: \(\$2,350,000\) should be 5% of \(\$47,000,000\)
5% of \(\$47,000,000 = 0.05 \times \$47,000,000 = \$2,350,000\) ✓

Final Answer

The amount of the increase was \(\$2,350,000\).

Looking at our answer choices, this matches choice B: \(\$2,350,000\).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "5 percent higher than" as a simple percentage calculation
Students often think that if 1980 profits are "5 percent higher than" 1979 profits, they can simply calculate 5% of \(\$49,350,000\) to get the increase. This fails because \(\$49,350,000\) is the NEW amount (105% of the original), not the BASE amount from 1979.

2. Confusing what represents 100% vs 105%
Students may incorrectly assume that the 1979 profits represent 105% and the 1980 profits represent 100%, reversing the relationship. This happens when they don't carefully track which year had the higher profits.

3. Setting up the wrong equation due to percentage language confusion
Instead of recognizing that 1980 profits = \(1.05 \times\) (1979 profits), students might set up: 1979 profits = \(1.05 \times\) (1980 profits), which leads to finding the wrong base amount.

Errors while executing the approach

1. Arithmetic errors when dividing by 1.05
Students often struggle with dividing \(\$49,350,000\) by 1.05, especially when converting to the fraction form (21/20). Common mistakes include incorrect decimal calculations or errors when multiplying by 20/21.

2. Rounding errors during intermediate calculations
When calculating \(\$49,350,000 \div 1.05\), students might round intermediate steps incorrectly, leading to a wrong value for 1979 profits and consequently an incorrect increase amount.

Errors while selecting the answer

1. Selecting the 1979 profits instead of the increase amount
After correctly calculating that 1979 profits were \(\$47,000,000\), students might mistakenly select this as their final answer, forgetting that the question asks specifically for "the amount of the increase."

2. Choosing 5% of the 1980 profits
Students who made the initial conceptual error might calculate 5% of \(\$49,350,000 = \$2,467,500\) and select answer choice C, not realizing this represents 5% of the wrong base amount.