Question

proportional reasoning (fractions and dec\
question\
the table below shows that the number\
of miles driven by jamal is directly\
proportional to the number of gallons\
he used.\
\

gallons used	miles driven	\
---	---	\
14	525	\
43	1612.5	\
47	1762.5	\

\
how many gallons of gas would he\
need to travel 296.25 miles?\
\
answer\
attempt 1 out of 2

Explanation:

Step1: Find the constant of proportionality

Since miles driven (\(y\)) is directly proportional to gallons used (\(x\)), the relationship is \(y = kx\), where \(k\) is the constant of proportionality. Using the first row: \(k=\frac{y}{x}=\frac{525}{14}\)
\(k = 37.5\)

Step2: Use the constant to find gallons for 296.25 miles

We know \(y = 296.25\) and \(y=kx\), so \(x=\frac{y}{k}\). Substitute \(y = 296.25\) and \(k = 37.5\):
\(x=\frac{296.25}{37.5}\)
\(x = 7.9\)? Wait, no, let's recalculate \(\frac{296.25}{37.5}\). 37.5×7 = 262.5, 296.25 - 262.5 = 33.75, 33.75/37.5 = 0.9, so 7 + 0.9 = 7.9? Wait, no, wait 37.5×7.9 = 37.5×(8 - 0.1)= 300 - 3.75 = 296.25. Wait, but let's check with another row. 43×37.5 = 1612.5, which matches. 47×37.5 = 1762.5, which also matches. So the formula is correct. Wait, but let's do the division properly: 296.25 ÷ 37.5. Multiply numerator and denominator by 1000 to eliminate decimals: 296250 ÷ 37500. Divide numerator and denominator by 750: 395 ÷ 50? Wait, no, 296250 ÷ 37500. Divide numerator and denominator by 100: 2962.5 ÷ 375. Divide by 75: 39.5 ÷ 5 = 7.9. Wait, but let's check again. Wait 37.5 × 7.9 = 37.5×7 + 37.5×0.9 = 262.5 + 33.75 = 296.25. Yes. So \(x = 7.9\)? Wait, but maybe I made a mistake in step 1? Wait 525 divided by 14: 14×37 = 518, 525 - 518 = 7, so 37 + 7/14 = 37.5. Yes, 14×37.5 = 525. Correct. Then 296.25 / 37.5: let's do this division as 296.25 ÷ 37.5. Move decimal two places: 29625 ÷ 3750. Divide numerator and denominator by 75: 395 ÷ 50 = 7.9. Wait, but maybe the problem expects an integer? Wait, no, let's check the numbers again. Wait 525/14 = 37.5, correct. 1612.5/43 = 37.5, correct. 1762.5/47 = 37.5, correct. So 296.25 / 37.5 = 7.9. Wait, but maybe I miscalculated. Wait 37.5 × 7.9 = 296.25, yes. So the number of gallons is 7.9? Wait, but the table has whole numbers for gallons, but the miles have decimals. So it's okay.

Wait, no, wait 296.25 divided by 37.5: 37.5 × 7 = 262.5, 296.25 - 262.5 = 33.75. 33.75 / 37.5 = 0.9. So 7 + 0.9 = 7.9. So the answer is 7.9? Wait, but maybe I made a mistake. Wait, let's check with another approach. Let's set up a proportion. Let \(x\) be the gallons needed for 296.25 miles. So \(\frac{525}{14}=\frac{296.25}{x}\). Cross - multiply: 525x = 14×296.25. 14×296.25 = 4147.5. Then x = 4147.5 / 525. 525×7 = 3675, 4147.5 - 3675 = 472.5, 472.5 / 525 = 0.9, so x = 7.9. Yes, that's correct.

Answer:

7.9