# One tank is filling at a rate of 5/8 gallon per 7/10 hour. A second tank is filling at rate of 5/9 gallon per 2/3 hour. Which tank is filling faster?

Nov 11, 2016

Tank A is filling faster.

#### Explanation:

You need to compare the rates in the same units to be able to decide which is better. Both are given as gallons in a fraction of an hour, but it would be better to find out how much each fills in ONE hour.

To get a RATE in 'Gallons per Hour',:

$\rightarrow$ divide the number of gallons by the time in hours.

TANK A$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times x}$ TANK B

$\frac{5}{8} \div \frac{7}{10} \textcolor{w h i t e}{\times \times \times \times \times \times \times \times x} \frac{5}{9} \div \frac{2}{3}$

=$\frac{5}{\cancel{8}} ^ 4 \times {\cancel{10}}^{5} / 7 \textcolor{w h i t e}{\times \times \times \times \times \times} = \frac{5}{\cancel{9}} ^ 3 \times \frac{\cancel{3}}{2}$

$= \frac{25}{28} \text{ } \textcolor{w h i t e}{\times \times \times \times \times \times \times \times} = \frac{5}{6}$

These are the two rates given in Gallons per Hour

A common denominator will be awkward to work with, but it is easy to make the numerators the same, and then compare.

$\frac{25}{28} \text{ and } \frac{5}{6} \times \frac{5}{5}$

$\frac{25}{28} \text{ and } \frac{25}{30}$

Note that $\frac{1}{28} > \frac{1}{30}$

Remember that. the bigger the denominator, the smaller the portion.

$\frac{25}{28} > \frac{25}{30}$

Tank A is filling faster.

[Decimals would have given the answer immediately, but they are not as much fun!]

$\frac{25}{28} = 0.89286 \text{ and } \frac{5}{6} = 0.833333$

Nov 13, 2016

Tank 1 at $\frac{5}{8}$ gallons per $\frac{7}{10}$ hour

#### Explanation:

The most straightforward way is to standardise the time and then just compare the rate of filling.

I chose to standardise the time to 1 hour.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Consider Tank 1 "-> 5/8" g per "7/10" hour}}$

To change $\frac{7}{10}$ hours into 1 hour multiply by $\textcolor{b l u e}{\frac{10}{7}}$

So we have color(green)(color(blue)(10/7)(5/8" gallons per "7/10" hour")_ giving:

$\textcolor{g r e e n}{\left(\textcolor{b l u e}{\frac{10}{7}} \times \frac{5}{8}\right) \text{ gallons per "(color(blue)(10/7)xx7/10)" hour}}$

color(brown)("Tank 1" ->25/28 " gallons per 1 hour"
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b r o w n}{\text{Consider Tank 2 "-> 5/9" gallons per "2/3" hour}}$

To change $\frac{2}{3}$ hours into 1 hour multiply by $\textcolor{b l u e}{\frac{3}{2}}$

so we have $\textcolor{g r e e n}{\textcolor{b l u e}{\frac{3}{2}} \left(\frac{5}{9} \text{ gallons per "2/3" hour}\right)}$

color(green)((color(blue)(3/2)xx5/9)" gallons per "(color(blue)(3/2xx)2/3" hour"))

$\textcolor{b r o w n}{\text{tank 2 "-> 5/6" gallons per 1 hour}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So for 1 hour the filling ratio is tank 1 : tank 2$\to \frac{25}{28} : \frac{5}{6}$

We need to make the denominators the same. Note that $6 \times 4 \frac{2}{3} = 28$

Multiply tank 2 by 1 but in the form of $1 = \frac{4 \frac{2}{3}}{4 \frac{2}{3}}$

So we are comparing $\frac{25}{28} : \left(\frac{5}{6} \times \frac{4 \frac{2}{3}}{4 \frac{2}{3}}\right)$

$\implies \text{tank 1 : tank 2 } \to \frac{25}{28} : \frac{23 \frac{1}{3}}{28}$

So tank 1 is filling faster