Okay, let's see. I need to figure out how many inches on the map correspond to 53.25 miles in real life. The problem says that 3 inches on the map equal 10 miles in reality. Hmm, so this is a scale conversion problem. I remember that scales can be converted using ratios or proportions. Let me think step by step.

First, the given ratio is 3 inches to 10 miles. So for every 10 miles, it's 3 inches on the map. I need to find out how many inches correspond to 53.25 miles. Maybe I can set up a proportion here. Let me denote the unknown distance on the map as x inches. Then, the proportion would be:

3 inches / 10 miles = x inches / 53.25 miles

Yes, that makes sense. If 3 inches is to 10 miles as x inches is to 53.25 miles, then solving for x should give me the answer. To solve the proportion, I can cross-multiply:

3 * 53.25 = 10 * x

Let me calculate 3 multiplied by 53.25. Let's do that. 53.25 times 3. Well, 50 times 3 is 150, and 3.25 times 3 is 9.75. So adding those together, 150 + 9.75 = 159.75. So, 3 * 53.25 = 159.75. Therefore, 159.75 = 10x. To solve for x, divide both sides by 10: x = 159.75 / 10 = 15.975 inches.

But the problem asks for the answer expressed as a fraction. Hmm, 15.975 inches. Let me convert that decimal to a fraction. 0.975 is equal to 975/1000. Let me simplify that fraction. Divide numerator and denominator by 25: 975 ÷ 25 = 39, 1000 ÷ 25 = 40. So 39/40. Therefore, 0.975 is 39/40. So 15.975 inches is 15 and 39/40 inches. To express this as an improper fraction, multiply the whole number by the denominator and add the numerator: 15 * 40 = 600, 600 + 39 = 639. So 639/40 inches. Let me check if that reduces further. 639 divided by 3 is 213, 40 divided by 3 is not an integer. So 639/40 is the simplified fraction. Let me verify if this is correct.

Wait, another way to approach this is to use unit rates. Since 3 inches correspond to 10 miles, then 1 inch corresponds to 10/3 miles. But we need the inverse. Wait, no. Wait, 3 inches = 10 miles. So 1 inch = 10/3 miles. But we need miles to inches. So maybe miles per inch. Wait, 10 miles per 3 inches. So miles per inch would be 10/3 miles per inch. But we need inches per mile. So inches = (3/10) miles. Wait, maybe I need to set up the conversion factor correctly.

Wait, perhaps it's better to use dimensional analysis. If 3 inches correspond to 10 miles, then the conversion factor is 3 inches / 10 miles. So to convert miles to inches, multiply by 3/10. So 53.25 miles * (3 inches / 10 miles) = 53.25 * 3 / 10 inches. Let's compute that. 53.25 * 3 = 159.75, then divided by 10 is 15.975 inches, which is the same as before. So that's 15.975 inches, which is 15 and 39/40 inches, or 639/40 as an improper fraction. So that seems consistent.

Alternatively, maybe I can think of it as setting up the ratio as 3/10 = x/53.25, which is the same proportion as before. Solving for x gives x = (3/10)*53.25. Which again is 3*53.25/10 = 159.75/10 = 15.975. So same result.

Let me check if 15.975 inches is indeed 639/40. 40*15 = 600, 639 - 600 = 39. So yes, 639/40 is 15 39/40, which is 15.975. So that's correct.

But just to make sure I didn't make any calculation errors. Let me recompute 3*53.25. 53.25 * 3. Let's break it down: 50*3 = 150, 3.25*3 = 9.75. 150 + 9.75 = 159.75. Then divide by 10: 15.975. Yes. So that's correct.

Alternatively, maybe using fractions all the way. Let's represent 53.25 as a fraction. 53.25 is equal to 53 and 1/4, which is 213/4. So 213/4 miles. Then, the scale is 3 inches per 10 miles. So inches = (3/10) * miles. Therefore, inches = (3/10)*(213/4) = (3*213)/(10*4) = 639/40. Which is the same as before. So that's another way to verify.

So 639 divided by 40 is 15.975, which is 15 and 39/40. So the answer is 639/40 inches. Since the problem asks for the answer expressed as a fraction, 639/40 is the correct form. Let me check if it can be simplified further. The numerator is 639 and denominator is 40. Prime factors of 40 are 2^3*5. Prime factors of 639: 639 divided by 3 is 213, 213 divided by 3 is 71. So 639 = 3*3*71. 71 is a prime number. 40 is 2^3*5. So no common factors between numerator and denominator. Therefore, 639/40 is in simplest terms.

Therefore, the distance between the buildings on the map is 639/40 inches.

**Final Answer**
The distance between the buildings on the map is \boxed{\dfrac{639}{40}} inches.
Therefore, the distance between the two buildings on the map is $\boxed{\frac{639}{40}}$ inches.
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