A speciality supermarket sells 320 products. Each of these products was either a cosmetic product or a nutrition product. Each of these products was also either a foreign product or a domestic product. Each of these products had at least one of the two approvals – FDA or EU.

The following facts are also known:

1. There were equal numbers of domestic and foreign products.
2. Half of the domestic products were FDA approved cosmetic products.
3. None of the foreign products had both the approvals, while 60 domestic products had both the approvals.
4. There were 140 nutrition products, half of them were foreign products.
5. There were 200 FDA approved products. 70 of them were foreign products and 120 of them were cosmetic products.

How many cosmetic products did not have FDA approval?

Question

A speciality supermarket sells 320 products. Each of these products was either a cosmetic product or a nutrition product. Each of these products was also either a foreign product or a domestic product. Each of these products had at least one of the two approvals – FDA or EU.

The following facts are also known:

1. There were equal numbers of domestic and foreign products.
2. Half of the domestic products were FDA approved cosmetic products.
3. None of the foreign products had both the approvals, while 60 domestic products had both the approvals.
4. There were 140 nutrition products, half of them were foreign products.
5. There were 200 FDA approved products. 70 of them were foreign products and 120 of them were cosmetic products.

How many cosmetic products did not have FDA approval?

Accepted Answer

From the given data:

1. **Total cosmetic products**:  
   - Domestic cosmetics = **90**.  
   - Foreign cosmetics = **90**.  
   - Total cosmetics = **180**.

2. **FDA-approved cosmetics**:  
   - Domestic FDA-approved cosmetics = **80**.  
   - Foreign FDA-approved cosmetics = **50**.  
   - Total FDA-approved cosmetics = **130**.

3. **Cosmetic products without FDA approval**:  
   Subtract the FDA-approved cosmetics from the total cosmetics:  
   $$
   	ext{Non-FDA approved cosmetics} = 	ext{Total cosmetics} - 	ext{FDA-approved cosmetics}.
   $$  
   $$
   = 180 - 130 = 50.
   $$

4. **Breakdown of non-FDA approved cosmetics**:  
   - Domestic non-FDA approved cosmetics = Total domestic cosmetics - Domestic FDA-approved cosmetics:  
     $$
     90 - 80 = 10.
     $$  
   - Foreign non-FDA approved cosmetics = Total foreign cosmetics - Foreign FDA-approved cosmetics:  
     $$
     90 - 50 = 40.
     $$

Thus, the number of **domestic non-FDA approved cosmetic products** is **10**, which matches Option C.

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