Difficulty: Easy
Correct Answer: 1.20
Explanation:
Introduction / Context:
This is a direct proportion question using the notation a : b :: c : d, which means a is to b as c is to d. The problem checks the ability to convert a verbal proportion into an equation and solve for an unknown variable using simple cross multiplication.
Given Data / Assumptions:
Concept / Approach:
In any proportion a : b :: c : d, we have a / b = c / d. Therefore, we equate the two fractions and solve for the unknown using cross multiplication. This is a fundamental technique for proportional reasoning and ratio problems.
Step-by-Step Solution:
Step 1: Translate the proportion into an equation: 0.75 / x = 5 / 8.
Step 2: Cross multiply to remove the denominators: 0.75 * 8 = 5 * x.
Step 3: Compute 0.75 * 8 = 6.0.
Step 4: So 6.0 = 5x.
Step 5: Divide both sides by 5 to get x = 6.0 / 5.
Step 6: Therefore, x = 1.2 which can be written as 1.20.
Verification / Alternative check:
Check the proportion with x = 1.2. We have 0.75 / 1.2 and 5 / 8. Calculate 0.75 / 1.2 = 0.625 and 5 / 8 = 0.625. Since both fractions are equal, the proportion holds exactly. Hence x = 1.20 is correct.
Why Other Options Are Wrong:
Values like 1.12, 1.16, 1.30 or 1.05 do not satisfy the proportion 0.75 : x :: 5 : 8. If you plug any of them into 0.75 / x, the result will not be equal to 5 / 8, so the fundamental proportional relationship fails. They are merely distractors based on likely arithmetic slips.
Common Pitfalls:
A frequent error is to invert the proportion incorrectly, for example by writing x / 0.75 = 5 / 8 and then solving, which gives a different value. Another mistake is to approximate intermediate values too early instead of keeping them exact. It is also easy to confuse the colon notation and write 0.75 : x = 5 : 8 without understanding that it really means 0.75 / x = 5 / 8.
Final Answer:
The value of x is 1.20.
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