Sure! Let's solve this step-by-step:
1. Identify the given ratio: The ratio of toffee:coffee:orange:mint chocolates is given as [tex]\(5:4:2:3\)[/tex]. We will denote these values as follows:
- Toffee = 5 parts
- Coffee = 4 parts
- Orange = 2 parts
- Mint = 3 parts
2. Calculate the total ratio sum excluding hazelnut chocolates: The total number of parts without considering hazelnut chocolates is:
[tex]\[
5 + 4 + 2 + 3 = 14 \text{ parts}
\][/tex]
3. Include the hazelnut chocolates: According to the problem, we should include hazelnut chocolates which will be 1 additional part. Therefore, the total number of parts becomes:
[tex]\[
14 + 1 = 15 \text{ total parts}
\][/tex]
4. Determine the total number of chocolates in the box: We are told that the probability of picking a hazelnut chocolate is [tex]\(\frac{1}{8}\)[/tex]. This means that out of every 8 chocolates, 1 is hazelnut. Therefore, the number of total chocolates can be determined by multiplying the total parts by 8:
[tex]\[
15 \times 8 = 120 \text{ chocolates}
\][/tex]
5. Calculate the number of hazelnut chocolates: Given the probability of [tex]\(\frac{1}{8}\)[/tex] for hazelnut chocolates, we can now find the number of hazelnut chocolates by dividing the total number of chocolates by 8:
[tex]\[
\frac{120}{8} = 15 \text{ hazelnut chocolates}
\][/tex]
So, there are [tex]\(15\)[/tex] hazelnut chocolates in the box.
