Sandy's orchard has only apple trees and pear trees and she is wanting to add some avocado trees. Currently, there are 4 apple trees for every 3 pear trees. She is wanting to plant enough avocado trees so that she has 2 avocado trees for every 5 pear trees in the orchard.

Question

Sandy is planting 18 avocado trees, making the ratio of avocado trees to apple trees be
blank to blank

Answer 1

Let's start by determining the current number of apple and pear trees in Sandy's orchard based on the given ratio.

We know that the ratio of apple trees to pear trees is 4 to 3. Let's denote the number of apple trees as $A$ and the number of pear trees as $P$. We can express this ratio mathematically as:

$$\frac{A}{P}=\frac{4}{3}$$

This means that for every 4 apple trees, there are 3 pear trees. We can express the number of apple trees in terms of the number of pear trees as:

$$A=\frac{4}{3}P$$

Next, Sandy is planting 18 avocado trees. She wants to maintain a ratio of 2 avocado trees for every 5 pear trees. Let's denote the number of avocado trees as $Av$. The relationship for the avocado trees in terms of the pear trees is:

$$\frac{Av}{P}=\frac{2}{5}$$

From this, we can express the number of avocado trees in terms of the number of pear trees:

$$Av=\frac{2}{5}P$$

Now, we know she is planting 18 avocado trees, so we can set up the equation:

$$18=\frac{2}{5}P$$

To solve for $P$, we can multiply both sides by 5:

$$90=2P$$

Next, divide both sides by 2:

$$P=45$$

Now that we have the number of pear trees, we can find the number of apple trees using our earlier equation:

$$A=\frac{4}{3}P=\frac{4}{3}\times 45=60$$

So, there are 60 apple trees and 45 pear trees.

Now we can find the ratio of avocado trees to apple trees. We have 18 avocado trees and 60 apple trees. The ratio can be expressed as:

$$\frac{Av}{A}=\frac{18}{60}$$

To simplify this ratio, we can divide both the numerator and the denominator by 6:

$$\frac{18÷6}{60÷6}=\frac{3}{10}$$

Thus, the ratio of avocado trees to apple trees is 3 to 10.