Which of the following is equal to the ratio of 5:4?

1. Translate the problem requirements: We need to find which expression equals the ratio \(5:4\), which can be written as \(\frac{5}{4}\) or \(1.25\)
2. Convert the target ratio to decimal form: Calculate \(5 \div 4 = 1.25\) to have a clear benchmark for comparison
3. Systematically evaluate each answer choice: Work through each option using basic arithmetic and exponent rules to see which equals \(1.25\)
4. Apply exponent simplification where needed: For expressions with powers, use the rule that \(\frac{\mathrm{a}^m}{\mathrm{a}^n} = \mathrm{a}^{(m-n)}\) to simplify before calculating

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for. The question asks for an expression that equals "the ratio of 5 to 4." In everyday language, this means we want to find which answer choice gives us the same result as dividing 5 by 4.

When we say "ratio of 5 to 4," we can write this as the fraction \(\frac{5}{4}\). This is our target - whatever expression we choose must equal \(\frac{5}{4}\).

Process Skill: TRANSLATE

2. Convert the target ratio to decimal form

To make our comparison easier, let's calculate what 5 divided by 4 actually equals:
\(5 \div 4 = 1.25\)

So any expression that equals \(1.25\) will be our correct answer. This decimal form will make it much easier to check our work as we go through each choice.

3. Systematically evaluate each answer choice

Now let's work through each option to see which one gives us \(1.25\):

Choice A: \(\frac{(0.8)^3}{(0.8)^4}\)
Before we calculate, let's think about what this means. We have the same base \((0.8)\) in both the numerator and denominator, just with different exponents.

Choice B: \(\frac{5^2}{4^2}\)
This means \(\frac{(5 \times 5)}{(4 \times 4)} = \frac{25}{16}\)
Let's calculate: \(25 \div 16 = 1.5625\)
This doesn't equal \(1.25\), so this isn't our answer.

Choice C: \(\frac{4}{5}\)
This is straightforward: \(4 \div 5 = 0.8\)
This doesn't equal \(1.25\), so this isn't our answer.

Choice D: \(\frac{(2)(5)}{2^4}\)
This means \(\frac{(2 \times 5)}{2^4} = \frac{10}{16} = \frac{5}{8}\)
Calculating: \(5 \div 8 = 0.625\)
This doesn't equal \(1.25\), so this isn't our answer.

Choice E: \(\frac{1.25}{0.4}\)
Let's calculate: \(1.25 \div 0.4 = 3.125\)
This doesn't equal \(1.25\), so this isn't our answer.

4. Apply exponent simplification where needed

Let's go back to Choice A and use our exponent rules to simplify it properly.

For \(\frac{(0.8)^3}{(0.8)^4}\), we can use the rule that when we divide powers with the same base, we subtract the exponents:
\(\frac{(0.8)^3}{(0.8)^4} = (0.8)^{3-4} = (0.8)^{-1}\)

Now, what does \((0.8)^{-1}\) mean? A negative exponent means we take the reciprocal:
\((0.8)^{-1} = \frac{1}{(0.8)} = 1 \div 0.8\)

Let's calculate this: \(1 \div 0.8 = 1.25\)

Perfect! This equals our target ratio of \(1.25\).

Final Answer

Choice A equals \(1.25\), which is exactly the same as our target ratio of 5 to 4.

To verify: \(\frac{(0.8)^3}{(0.8)^4} = (0.8)^{-1} = \frac{1}{0.8} = 1.25 = \frac{5}{4}\)

The correct answer is A.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "ratio of 5 to 4"
Students may confuse the order and think "ratio of 5 to 4" means \(\frac{4}{5}\) instead of \(\frac{5}{4}\). This fundamental misunderstanding would lead them to look for expressions that equal \(0.8\) instead of \(1.25\), potentially selecting Choice C \((\frac{4}{5}))\) as their answer.

2. Not converting to a common form for comparison
Students might try to work with fractions throughout without converting \(\frac{5}{4}\) to its decimal equivalent of \(1.25\). This makes it much harder to evaluate choices like A and E efficiently, leading to more time-consuming calculations and potential errors.

Errors while executing the approach

1. Incorrect application of exponent rules
When evaluating Choice A, students may incorrectly apply the quotient rule for exponents. Instead of \(\frac{(0.8)^3}{(0.8)^4} = (0.8)^{3-4} = (0.8)^{-1}\), they might add the exponents or make other mistakes, leading to an incorrect simplification.

2. Mishandling negative exponents
Even if students correctly get \((0.8)^{-1}\), they may not properly convert the negative exponent to a reciprocal. They might think \((0.8)^{-1} = -0.8\) instead of \(\frac{1}{0.8}\), leading to an incorrect final value.

3. Arithmetic errors in decimal division
Students may make calculation mistakes when computing divisions like \(1 \div 0.8 = 1.25\) or \(1.25 \div 0.4 = 3.125\). These computational errors can lead them to incorrectly eliminate the right answer or incorrectly accept a wrong answer.

Errors while selecting the answer

No likely faltering points - once students have correctly calculated that Choice A equals \(1.25\) and matches the target ratio of \(\frac{5}{4}\), the answer selection is straightforward.