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Question: A gerbil drinks 2/3 of a water bottle every 1.5 days. Use a unit rate to discover how many water bottles the gerbil drinks in 5 days.

When given problems like this before, I have used whole numbers and was easily able to figure out a unit rate for 1 day. Since I have fractions, I am confused on how to break 1.5 down to 1 all while changing the 2/3.

When I did this problem without a unit rate, I multiplied 2/3 by 3 and discovered that the gerbil drank 2 water bottles in 4.5 days. I then took 1/2 of 2/3 and got that the gerbil drinks around 2.3 water bottles in 5 days!

How do I compute this using unit rates?

Chloe
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2 Answers2

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So we are given that the gerbil drinks $\frac{2}{3}$ bottles every $\frac{3}{2}$ days.

Hence, the amount it drinks in one day is $\frac{2}{3} \div \frac{3}{2} = \frac{4}{9}$ bottles.

Hence, the amount it drinks in five days is $\frac{4}{9} \times 5 = \frac{20}{9} = 2.\overline{2}$

The second step is the most important here: all you are doing is simple division, that gives you the unit rate.

Doing it your way, you said that $2$ liters are consumed in $4.5$ days. The remaining is $\frac{1}{3}$ of $\frac{3}{2}$ days, so the amount consumed in that time is $\frac{1}{3}*\frac{2}{3} = \frac{2}{9}$. Hence the answer is $2\frac{2}{9} = 2.\overline{2}$

By the way, feel free to clear doubts, I know it's a sensitive topic.

Sarvesh Ravichandran Iyer
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  • quick question related to this post: when we say something like “five meters per three seconds”, is this equivalent to saying “five-thirds meters per second” by converting to a unit rate? – Taylor Rendon Jul 17 '22 at 15:02
  • @TaylorRendon Yes, with the context I assume you have, you are correct. Let me know if you wish to expand upon your doubt in more detail, and we can discuss it in this thread. – Sarvesh Ravichandran Iyer Jul 17 '22 at 15:05
  • Wonderful, to clarify here is my question with further context: suppose the slope of a line is $m = \frac{5}{3}$ (measured in meters/seconds), then is saying “the rate of change is five meters per three seconds” equivalent to saying “the rate of change is five-thirds meters per second”? – Taylor Rendon Jul 17 '22 at 15:46
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    @TaylorRendon Yes, this is perfectly correct. This is what I would consider an example of the "unit rate" method. The point is, if you're covering $5$ meters in $3$ seconds, then the "unit rate" method makes you ask the question "what is the distance covered in unit time?" first, and this quantity is the same as the rate (and is obtained by division, like you and I did). For example, $5$ divided by $3$ i.e. $5/3$ would be the "unit rate" : in other words, "the rate of change is five-thirds meters per second". If You're attempting the unit rate method, then your first step is appropriate. – Sarvesh Ravichandran Iyer Jul 17 '22 at 16:13
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Simply write down an expression for the rate and extend the fraction so that the denominator is $1$:

The rate given is $\frac{2}{3} \text{ bottles per }\frac{3}{2} \text{ days.}$ Here "per" means "divided by", so we have $$\frac{\frac{2}{3}}{\frac{3}{2}}\text{ bottles per day}=\frac{4}{9} \text{ bottles per day} \implies 5\times \frac{4}{9} \text{ bottles per $5$ days}=\frac{20}{9} \text{ bottles per $5$ days}$$

Bobson Dugnutt
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