Solve the equation 0.78x = 50.

Step-by-step solution

1. 1

   Divide both sides by the coefficient. The variable is multiplied by $0.78$, so dividing undoes it:

   $x=\frac{50}{0.78}$

   Stopping here and typing it into a calculator gives a long decimal; converting to a fraction first keeps the answer exact.

2. 2

   Convert the decimal coefficient to a fraction. Two decimal places means hundredths, and the fraction reduces:

   $0.78=\frac{78}{100}=\frac{39}{50}$

   (Dividing top and bottom by $2$; note $39=3\times13$ and $50=2\times5^{2}$ share no factor, so this is lowest terms.)

3. 3

   Divide by the fraction — multiply by its reciprocal.

   $x=50\div\frac{39}{50}=50\cdot\frac{50}{39}=\frac{2500}{39}$

   Since $2500=2^{2}\cdot5^{4}$ shares no factor with $39=3\cdot13$, the fraction $\tfrac{2500}{39}$ is already in lowest terms.

4. 4

   Give the decimal form. Because the denominator $39$ contains primes other than $2$ and $5$, the decimal repeats:

   $x=\frac{2500}{39}=64.\overline{102564}\approx64.103$

   The exact fraction is the honest answer; $64.1$ is a rounding.

5. 5

   Verify in the original equation. $\dfrac{2500}{39}\times\dfrac{39}{50}=\dfrac{2500}{50}=50$ ✓ — the $39$s cancel exactly, confirmed in exact rational arithmetic. As a rough check, $0.78\times64=49.92$, just under $50$, so a value slightly above $64$ is right ✓.