Question

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?

Answer 1

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',:

`rarr` divide the number of gallons by the time in hours.

TANK A`color(white)(xxxxxxxxxxxxxxxxx)` TANK B

`5/8 div 7/10color(white)(xxxxxxxxxxxxxxxxx)5/9 div 2/3`

=`5/cancel8^4 xx cancel10^5/7color(white)(xxxxxxxxxxxx)=5/cancel9^3 xx cancel3/2`

`=25/28" "color(white)(xxxxxxxxxxxxxxxx)=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.

`25/28" and "5/6 xx5/5`

`25/28" and "25/30`

Note that `1/28 > 1/30`

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

`25/28 > 25/30`

Tank A is filling faster.

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

`25/28 =0.89286 " and "5/6 = 0.833333`

Answer 2

Tank 1 at `5/8` gallons per `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.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Consider Tank 1 "-> 5/8" g per "7/10" hour")`

To change `7/10` hours into 1 hour multiply by `color(blue)(10/7)`

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

`color(green)((color(blue)(10/7)xx5/8)" gallons per "(color(blue)(10/7)xx7/10)" hour")`

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

`color(brown)("Consider Tank 2 "-> 5/9" gallons per "2/3" hour")`

To change `2/3` hours into 1 hour multiply by `color(blue)(3/2)`

so we have `color(green)(color(blue)(3/2)(5/9" gallons per "2/3" hour"))`

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

`color(brown)("tank 2 "-> 5/6" gallons per 1 hour")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So for 1 hour the filling ratio is tank 1 : tank 2`-> 25/28 : 5/6`

We need to make the denominators the same. Note that `6xx4 2/3=28`

Multiply tank 2 by 1 but in the form of `1=(4 2/3)/(4 2/3)`

So we are comparing `25/28 : (5/6xx (4 2/3)/(4 2/3))`

`=>"tank 1 : tank 2 "-> 25/28: (23 1/3)/28`

So tank 1 is filling faster