Question

If $ \dfrac{{241}}{{4000}} = \dfrac{{241}}{{{2^m} \times {5^n}}} $ , find the values of m and n, where m and n are non negative integers. Hence write its decimal expansion without actual division.

Answer 1

Hint: Prime factorize 4000, we will get a result in 2’s and 5’s raised to some powers. The exponents of 2 and 5 will give the values of m and n. Decimal Expansion is the form of a number that has a decimal point in it. Convert 241/4000 into decimal.

Complete step-by-step answer:
We are given $ \dfrac{{241}}{{4000}} = \dfrac{{241}}{{{2^m} \times {5^n}}} $ and we have to find the values of m and n which are the exponents of 2 and 5 respectively; m and n are non negative integers.
Prime factorize 4000.
4000 can be written as two times of 2000.
 $ 4000 = 2 \times 2000 $ s
2000 can be written as two times of 1000.
 $ 4000 = 2 \times 2 \times 1000 $
1000 can be written as two times of 500.
 $ 4000 = 2 \times 2 \times 2 \times 500 $
500 can be written as two times of 250.
 $ 4000 = 2 \times 2 \times 2 \times 2 \times 250 $
250 can be written as two times of 125.
 $ 4000 = 2 \times 2 \times 2 \times 2 \times 2 \times 125 $
125 can be written as five times of 25.
 $ 4000 = 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 25 $
25 can be written as five times of 5.
 $
  4000 = 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 \\
  4000 = {2^5} \times {5^3} \\
  $
Compare the above result with the question.
 $
  {2^m} \times {5^n} = {2^5} \times {5^3} \\
   \to m = 5,n = 3 \\
  $
The value of m is 5, n is 3.
Write $ \dfrac{{241}}{{4000}} $ in decimal expansion.
 $
  \dfrac{{241}}{{4000}} = \dfrac{{241}}{{4 \times 1000}} \\
   = \dfrac{{60.25}}{{1000}} \\
   = 0.06025 \\
  $
 $ \dfrac{{241}}{{4000}} $ in decimal expansion is 0.06025.

Note: :Prime Factorization is finding the prime numbers which are factors which multiplied together will result in the original number. In the decimal system, each decimal place consists of a digit arranged such that each digit is multiplied by a power (negative powers) of 10, decreasing from left to right. Decimal expansion includes a decimal point and writing the numbers in forms of decimals.