Problems based on number system (Part - 1)

Problems based on number system

1). Express 0.47777.. in the form of p/q 2). Express 0.001001.. in the form of p/q

Ans. Let X = 0.47777... Ans. Let X = 0.00101.0.. ......(1)

Multiply by 10 in both side Multiply by 1000 in both side

10X = 4.7777... ............ (1) 1000 X = 1.001001...

Multiply by 10 again in both side 1000 X = 1 +0.001001....

100X = 47.7777... 1000 X = 1 + X

100 X = 43 + 4.7777.... 1000 X - X = 1

100 X = 43 + 10 X 999 X = 1

100 X - 10 X = 43 X = 1/999

90 X = 43

X = 43/90

3). Express 0.6666.. in the form of p/q 4). Express 0.9999.. in the form of p/q

Ans. Let X = 0.6666... Ans. Let X = 0.9999... ......(1)

Multiply by 10 in both side Multiply by 10 in both side

10X = 6.666... 10X = 9.9999...

10 X = 6 + 0.6666.... 10 X = 9 + 0.9999....

10 X = 6 + X 10 X = 9 + X

10 X - X = 6 10 X - X = 9

9 X = 6 9 X = 9

X = 6/9 X = 9/9

X = 2/3 X = 1

5). Express 1.6666.. in the form of p/q 6). Express 0.5454.. in the form of p/q

Ans. Let X = 1.6666... Ans. Let X = 0.5454... ......(1)

Multiply by 10 in both side Multiply by 100 in both side

10X = 16.666... 100X = 54.5454...

10 X = 15 + 1.6666.... 100 X = 54 + 0.5454....

10 X = 15 + X 100 X = 54 + X

10 X - X = 15 100 X - X = 54

9 X = 15 99 X = 54

X = 15/9 X = 54/99

X = 5/3 X = 6/11

7). Express 0.3222.. in the form of p/q 8). Express 0.12333.. in the form of p/q

Ans. Let X = 0.322... Ans. Let X = 0.12333... ......(1)

Multiply by 10 in both side Multiply by 100 in both side

10X = 3.222... ...(1) 100X = 12.3333...

100 X = 32.22... + 3.222.... 1000 X = 123.3333....

100 X = 29 + 10X 1000 X = 111 + 12.333...

100 X - 10X = 29 1000 X = 111 + 100X

90 X = 29 1000X - 100X = 111

X = 29/90 900X = 111

X = 111/900
X = 37/300

9). Find the maximum number of digits in repeating block of 1/17

Sol. In the decimal expansions of 1/17, the maximum number of digits in repeating block is 16.
In actual division 1/17 = 0.0588235294...
So the maximum number of digits in repeating block is 16.

10). Three non- terminating and non-recurring decimals.
a). 0.4141...
b). 2.01001000100001.....
c). 3.1416... that is also known as pie

Milan Tomic

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