The table below shows the ratio of the number of employees registered for eye checkup to those registered for dental checkup in four different slots (A, B, C, and D) in a company.

| Time Slots | Ratio of Eye Checkup : Dental Checkup |
|------------|---------------------------------------|
| A | 5 : 4 |
| B | 7 : 3 |
| C | 11 : 4 |
| D | 1 : 5 |

Note: Employees registered are either for eye checkup or dental checkup.

If the total number of employees registered for eye checkup in slots A and D is 29, and the sum of employees registered for dental checkup in the same slots is 40, what is the difference between the total number of employees in slot A and slot D?

Question

The table below shows the ratio of the number of employees registered for eye checkup to those registered for dental checkup in four different slots (A, B, C, and D) in a company.

| Time Slots | Ratio of Eye Checkup : Dental Checkup |
|------------|---------------------------------------|
| A          | 5 : 4                                 |
| B          | 7 : 3                                 |
| C          | 11 : 4                                |
| D          | 1 : 5                                 |

Note: Employees registered are either for eye checkup or dental checkup.

If the total number of employees registered for eye checkup in slots A and D is 29, and the sum of employees registered for dental checkup in the same slots is 40, what is the difference between the total number of employees in slot A and slot D?

Accepted Answer

**Solution:**

1. Let the number of employees registered for eye checkup in slot A be $5x$ and for dental checkup be $4x$ (from ratio 5:4).

2. Let the number of employees registered for eye checkup in slot D be $y$ and for dental checkup be $5y$ (from ratio 1:5).

3. According to the problem, total eye checkup employees in slots A and D is 29:
   $$5x + y = 29 	ag{1}$$

4. Total dental checkup employees in slots A and D is 40:
   $$4x + 5y = 40 	ag{2}$$

5. From equation (1), express $y$ in terms of $x$:
   $$y = 29 - 5x$$

6. Substitute $y$ into equation (2):
   $$4x + 5(29 - 5x) = 40$$
   $$4x + 145 - 25x = 40$$
   $$-21x + 145 = 40$$
   $$-21x = 40 - 145 = -105$$
   $$x = \frac{105}{21} = 5$$

7. Find $y$:
   $$y = 29 - 5 	imes 5 = 29 - 25 = 4$$

8. Total employees in slot A:
   $$5x + 4x = 9x = 9 	imes 5 = 45$$

9. Total employees in slot D:
   $$y + 5y = 6y = 6 	imes 4 = 24$$

10. Difference between total employees in slots A and D:
    $$45 - 24 = 21$$

**Answer:** 21

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