In a decimal number, a bar over one or more consecutive digits means

Of all the answer choices, only \(\frac{91}{12}\) has a quotient of 7

Answer is (E)

Solution:

Let x = 7.58333… Then, 100x = 758.333… and 1000x = 7583.333… . If we subtract 100x from 1000x, the decimal parts cancel each other out and we are left with:

900x = 7583 - 758

900x = 6825

x = 6825/900 = 91/12

Alternate Solution:

When a decimal can be converted to a fraction, it’s either a finite decimal (e.g., 1.2, 3.45) or a repeating decimal (e.g., 0.666..., 0.07878…). We know how to convert a finite decimal to fraction (e.g., 1.2 = 12/10 and 3.45 = 345/100). We can do the same for a repeating decimal, but instead of “over 10”, “over 100,” it’s “over 9”, “over 99” and so on. Of course, that depends on the number of digits in the “block of digits” that are repeating. For example, in 0.666…, the block of the digits that is repeating is only 6. So the block is 6 and there is only 1 digit repeating. To convert the decimal to a fraction, we put the block over the number the number 9 and we have 0.666... = 6/9 = 2/3. Let’s have another example, in 0.375375...., the block is 375 and the number of digits repeating in this block is 3. Therefore, we are going to put 375 over 999. In other words, 0.375375… = 375/999. However, this method is only true when the block that is repeating is right after the decimal point. It will not be true if the decimal is 0.07878… or in our example, 7.58333…. On the other hand, we can modify our rule to make it work for decimals such as 0.07878… and 0.00375375…, i.e., decimals with all zeros (including the one to the left of the decimal point) before the repeating block. We will tweak the rule as follows:

Again count the number of digits in the block that are repeating (let’s say it’s m) and also count the number of zeros between the decimal point and the block (let’s say it’s n). Then the denominator of the fraction (before reducing) is the integer with m nines followed by n zeros. For example, if m = 2 and n = 1, put the block over 990. If m = 3 and n = 2, put the block over 99900 and so on. Therefore, 0.07878… = 78/990 and 0.00375375… = 375/99900.

At this point, you might say we still haven’t answered how to convert the decimal 7.58333… to a fraction since the block that is repeating, 3, is not preceded by 0s. Yes, it’s true, however, for such a decimal, we can always express it as a sum of a finite decimal and a repeating decimal that we’ve discussed. Notice that 7.58333… = 7.58 + 0.00333… Therefore, we have:

7.58333… = 7.58 + 0.00333… = 758/100 + 3/900 = 758/100 + 1/300 = 2274/300 + 1/300 = 2275/300 = 91/12

Answer: E

I think we can also approximate the answer which will be a bit quicker and is still sufficiently accurate for this question.

We can see that every answer choice has a 91 in the nominator.

If we move the decimal point of 7.583 one position to the right and make it 75.83/10 we know that the delta in order to get 91 in the nominator is approximately 15.

We therefore need two additional pieces of 7.583 to make it 91. The denominator would then be 12.

Answer E.

Please correct me if I am wrong in my reasoning. Thanks a lot.

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