Solving one-step problems - TACHS Math

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Question

Tim can ride his bike eight miles in one hour. How long will it take him to ride forty miles?

Answer

If Tim can ride his bike eight miles in one hour then we can write the following ratio:

$\frac{8\textup{ miles}}{1\textup{ hour}}$

Next, we want to know how long it will take him to ride forty miles. We can write the following ratio:

$\frac{40\textup{ miles}}{x\textup{ hours}}$

Next, we can set these equal to one another and form a proportion.

$\frac{8\textup{ miles}}{1\textup{ hour}}=\frac{40\textup{ miles}}{x\textup{ hours}}$

Cross multiply and solve for the variable.

$\frac{8\textup{ miles}}{1\textup{ hour}}\times\frac{40\textup{ miles}}{x\textup{ hours}}$

Simplify.

$8\times x=40\times1$

Divide each side of the equation by .

$\frac{8x}{8}=\frac{40}{8}$

Solve.

$x=5\textup{ hours}$

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Question

Barry, Jerry, and Mary order a pizza. Barry pays half the cost; Jerry pays one third of the cost; Mary pays the rest, which is $3. How much did the pizza cost?

Answer

Let be the cost of the pizza. Barry paid half the cost of the pizza, which is $\frac{1}{2}C$ ; Jerry paid one third of the cost, which is $\frac{1}{3}C$ , and Mary paid $3. The sum of their shares of the cost is equal to the total cost, so the equation to set up is

$C =\frac{1}{2}C+ \frac{1}{3}C + 3$

Solve for by isolating the variable at left. First, at right, collect the like terms by adding the coefficients of . This requires finding the least common denominator of the two fractions, which is 6:

$C =\frac{1 \cdot 3}{2 \cdot 3}C+ \frac{1 \cdot 2}{3 \cdot 2}C + 3$

$C =\frac{ 3}{6}C+ \frac{2}{6}C + 3$

$C =\left (\frac{ 3}{6} + \frac{2}{6} \right )C + 3$

$C = \frac{5}{6} C + 3$

Subtract $\frac{5}{6} C$ from both sides, again by collecting like terms:

$C - \frac{5}{6}C = \frac{5}{6} C + 3 - \frac{5}{6}C$

$\frac{6}{6} C - \frac{5}{6}C = 3$

$\left (\frac{6}{6} - \frac{5}{6} \right )C = 3$

$\frac{1}{6} C = 3$

Now multiply both sides by 6 to reverse multiplication by $\frac{1}{6}$ :

$6\cdot \frac{1}{6} C =6 \cdot 3$

The pizza cost $18.

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