Recall how to rewrite a repeating decimal as a fraction. Use long division to write the answer as a repeating decimal.
4930.672 kilograms per cubic meter
Practice makes perfect
We want to find the density of iodine, given the density of acetone. We will rewrite the given repeating decimal density of aceton as a fraction, then use it to find the density of iodine. Then we will rewrite our answer as a repeating decimal. First, let's recall how to write a repeating decimal as a fraction.
Write the equation x=d, where d is a repeating decimal.
Subtract the equation from the previous step from the equation 10^n x=10^n d, where n is the number of repeating digits.
Solve the equation for x.
We can use this process to rewrite 6.281 as a fraction. We will substitute 6.281 for d and 1 for n, as there is only one repeating digit in the decimal.
ccc x=d & ⇔ & x= 6.281 10^nx=10^n d & ⇔ & 10^1x=10^1( 6.281)
Let's simplify the second equation.
10^1x=10^1(6.281) ⇒ 10x=62.81
Now we are ready to subtract the first equation from the second one.
c r c l& 10x & = & 62.81 -( & x & = & 6.281 ) & 9x & = & 56.53
Finally, we can solve for x. Let's do it!
The density of iodine is about 5653900 times the density of acetone. From the exercise, we know that the density of acetone is 785 kilograms per cubic meter. We can find the density of iodine if we multiply 785 by 5653900.
785* 5653/900=4 437 605/900
Finally, we need to write this fraction as a repeating decimal. Let's use long division to divide 4 437 605 by 900.
The quotient begins to repeat after the third step. This means we can stop and write the fraction as a repeating decimal.
4 437 605/900=4930.672
The density of iodine is 4930.672 kilograms per cubic meter.
