Lawn mowing problem solving

Question

Kate can mow the lawn in 45 minutes. Kate's sister takes twice as long to mow the same lawn. If they both have a mower and mow the lawn together, how many minutes will it take them?

I know the answer is 30, but how?

Answer 1

This is about getting a "common denominator" in terms of how many lawns each of them can mow in the some common amount of time.

We are given that Kate can mow 1 lawn in 45 minutes, and her sister can mow 1 lawn in 90 minutes. So, we can see Kate can mow 2 lawns in 90 minutes (twice as many as her sister). Together this means they can mow 3 lawns in 90 minutes.

Finally, 3 lawns / 90 minutes = 1 lawn / 30 minutes.

Answer 2

We will define a rate equation as follows $$L=tR, $$ where $L$ is the number of lawns mowed, $t$ is the time is takes to mow said lawn (in minutes), and $R$ is the rate at which a lawn (or a fraction of a lawn) is mowed per minute.

From your question, we know the rate equation for Kate will be $$1=45R_K. $$ We find the rate at which Kate mows is $R_K=\frac{1}{45}\frac{\text{lawns}}{\text{min}}.$

We proceed in a similar fashion for Kate's sister. You said she could mow the same lawn in twice the time of Kate. This implies a rate equation of $$1=90R_S,$$ where $R_S=\frac{1}{90}\frac{\text{lawns}}{\text{min}}.$

Now, we must use a rate equation that combines the mowing rates for both Kate and her sister. This is given by $$L=tR_k+tR_S $$ or $$L=t(R_K+R_S). $$ We want the amount of time it taks to mow ONE lawn, therefore $L=1$. We also know the values for $R_K$ and $R_S$. We obtain the equation $$1=t\bigg(\frac{1}{45}+\frac{1}{90}\bigg). $$ Solving for $t$ yields the solution of $$t=30\ \text{minutes}. $$

Answer 3

Let's call Kate's sister Sue.

Sue takes twice as long as Kate to mow the lawn. This of course means that Kate mows the lawn twice as fast as Kate, so if they cooperate, Kate will cover twice the area of the lawn that Sue covers, i.e. Kate covers 2/3 of the lawn, while Sue covers 1/3.

But it takes Kate 2/3 * 45 minutes = 30 minutes to cover 2/3 of the lawn.

Answer 4

Kate mows like two Sues (Sue is Kate’s sister, who takes twice as long as Kate to mow the lawn), so together, it’s like having three Sues — they’ll mow the lawn in one-third the time as one Sue, so in 1/3 times 90 minutes, or 30 minutes.

Answer 5

Here's a formula for these types of questions, even when it's not double the time: $t=xy÷(x+y)$ with $t$ being the time/final answer, $x$ being how long Kate, in this instance, takes to do it and $y$ being how long Kate's sister takes to do it.