If the triangles are similar, what is the value of x?

9514 1404 393

Step-by-step explanation:

You can do these conversions the way shown in your attachment:

2) x = 0.18_8

  10x = 1.88_8

  9x = 1.88_8 -0.18_1 = 1.7

  x = 1.7/9 = 17/90

Or, you can recognize that 0.88_8 is 8/9, but 0.088_8 is that value divided by 10: 8/90. Then 0.18_8 is 0.1 + 0.088_8 = 1/10 + 8/90 = (9+8)/90 = 17/90

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Here are the others:

1) 0.5_5 = 5/9

3) 0.26_6 = 1/5 + 6/90 = (18 +6)/90 = 24/90 = 12/45

4) 0.287_87 = 1/5 + 87/990 = (198+87)/990 = 19/66

Further development

You have probably figured out that a 1-digit repeat that starts at the decimal point is expressed as a fraction with that digit divided by 9.

  x = 0.dd_d

  10x d.dd_d

  9x = d

  x = d/9 . . . . . . for any single-digit repeat

Now doubt, you're used to seeing 1/3 = 0.33_3 = 3/9, and 2/3 = 0.66_6 = 6/9.

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Similarly, a 2-digit repeat is the fraction with those two digits divided by 99 (an equal number of 9s). This holds for all n-digit repeats. The "raw" divisor is 10^n -1. In many cases, the fraction can be reduced, as in ...

  0.142857_142857 = 142857/999999 = 1/7

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When the repeating part of the decimal does not start at the decimal point, then the divisor (some number of 9s) must be adjusted by an appropriate power of 10. The non-repeating part can be expressed as a fraction in the usual way, then added to the value of the repeating part. Or, you can do the several-step process you show. Both work just fine.

About notation

Overbars are difficult to show in plain text. Not all methods of creating overbars are rendered properly in documents that show plain text. In typeset text, they are used to show the repeating digits. So, other methods are sometimes used in plain text to show the repeating digits. Here are a few:

  0.166_6 . . . . shows the 6 repeats

  0.16(6) . . . . . shows the 6 repeats [this notation is also used to indicate error limits for the number, so can be ambiguous]

  0.16... (repeating 6) . . . . . shows the 6 repeats