sara is thinking of a number that is a repeating deciaml of the form 0. abcdabcd...find sara's number using the following clues.
a. the digit that is located 100 places after the deciaml is 4.
b. the digit that is 150 places after the deciaml is 5.
c. the digit that is 175 places after the deciaml is 3
d. a+b=c+d
100 places after the decimal is the same as 4 places after the decimal.
100/25 = 4, so d = 4
150 places after the decimal is the same as 2 places after the decimal.
150 = 100/37 + 2, so b = 5
175 places after the decimal is the same as 3 places after the decimal.
c = 3
a + 5 = 3 + 4
a = 2
The repeating decimal is 0.25342534...
I assume you know how to convert that to a fraction that is a ratio of integers.
x = 2534/9999
0.253425343
To find Sara's number, we can make use of the given clues and solve the problem step by step.
Let's first analyze the pattern of the repeating decimal "0.abcdabcd...".
Since the digit that is located 100 places after the decimal is 4, we can say that the digit "a" (the first digit after the decimal point) is also 4.
Similarly, since the digit that is located 150 places after the decimal is 5, we can say that the digit "b" (the second digit after the decimal point) is 5.
Now, let's analyze the clue c. The digit that is located 175 places after the decimal is 3. Since the decimal repeats after every 4 digits, we can conclude that the 175th digit is equivalent to the 175 modulo 4 (remainder when divided by 4), which is 3. Therefore, the digit "c" is also 3.
The last clue, a+b=c+d, states that the sum of the digits "a" and "b" is equal to the sum of the digits "c" and "d". We already know the values of "a" and "c", so we can solve for "d".
a + b = c + d
4 + 5 = 3 + d
9 = 3 + d
To find the value of "d", subtract 3 from both sides of the equation:
9 - 3 = d
6 = d
So, the value of "d" is 6.
Now we have all the digits of Sara's number:
a = 4
b = 5
c = 3
d = 6
Therefore, Sara's number is 0.4536.