Question Video: Calculating Density to Compare Populations Mathematics • 6th Grade

Daniel and Charlotte both have a keen interest in gardening and are concerned by the number of slugs that they keep finding in their respective vegetable patches. They want to compare the number of slugs in each of their gardens but, due to the differences in size of their vegetable patches, they decide to compare the number of slugs per square foot. Daniel’s vegetable patch is a rectangle with dimensions 5 ft by 3 ft and Charlotte’s is a circular patch with a radius of 3 ft. One Saturday morning, Daniel counts 21 slugs in his entire vegetable patch and Charlotte counts 36. Work out the density of slugs in Daniel’s vegetable patch. Work out the density of slugs in Charlotte’s vegetable patch. Who has the more severe slug problem?

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Video Transcript

Daniel and Charlotte both have a keen interest in gardening and are concerned by the number of slugs that they keep finding in their respective vegetable patches. They want to compare the number of slugs in each of their gardens. But due to the differences in size of their vegetable patches, they decide to compare the number of slugs per square foot. Daniel’s vegetable patch is a rectangle with dimensions five foot by three foot. And Charlotte’s is a circular patch with a radius of three foot. One Saturday morning, Daniel counts 21 slugs in his entire vegetable patch and Charlotte counts 36. There are three parts to this question. Work out the density of slugs in Daniel’s vegetable patch. Work out the density of slugs in Charlotte’s vegetable patch. Who has the more severe slug problem?

We’re told that Daniel’s patch is rectangular and measures five foot by three foot, whereas Charlotte’s is circular with a radius of three foot. There were 21 slugs in Daniel’s vegetable patch and 36 in Charlotte’s. We will now clear some space to calculate the number of slugs they had per square foot. Let’s consider Daniel’s vegetable patch first. His patch was rectangular with dimensions three foot and five foot, and he found 21 slugs in his patch. We can calculate the area of any rectangle by multiplying the length by the width. In this case, we need to multiply five by three. This is equal to 15. Therefore, Daniel’s patch has an area of 15 square foot.

In order to calculate the density per square foot, we can, firstly, write the ratio of the area to the number of slugs. This is equal to 15 to 21. To calculate the density of slugs in Daniel’s patch, we need to calculate the unit ratio, how many slugs there are per square foot. This is written in the form one to 𝑛. We divide both sides of the ratio by 15, giving us the ratio one to 1.4. The density of slugs in Daniel’s vegetable patch is therefore 1.4 slugs per square foot.

We can now repeat this process for Charlotte. Charlotte’s vegetable patch was circular and had a ratio [radius] of three foot. She found 36 slugs in her vegetable patch. The area of any circle can be calculated by multiplying 𝜋 by the radius squared. In this question, this is equal to 𝜋 multiplied by three squared. This is equal to 28.2743 and so on. This means that the area of Charlotte’s vegetable patch is 28.27 square feet. For the purposes of this question, we will keep this as nine 𝜋. The ratio of area to slugs for Charlotte is therefore nine 𝜋 to 36. To find the unit ratio or density, we can divide both sides by nine 𝜋. 36 divided by nine 𝜋 is equal to 1.273 and so on. Rounding this to one decimal place gives us 1.3 slugs per square foot.

The three correct answers are 1.4, 1.3, and Daniel. As 1.4 is greater than 1.3, Daniel has the more severe slug problem.

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