The table below shows the total number of watches (Analog + Digital) sold by five shopkeepers on Monday along with the ratio of analog to digital watches sold by each. Shopkeeper D sells only two brands of watches: Fossil and Titan. If $ $\frac{5}{13} $$ of the analog watches and $ $\frac{6}{19} $$ of the digital watches sold by D are Fossil brand, find the total number of Titan watches sold by shopkeeper D on Monday.

| Shopkeepers | Total Sold Watches | Analog : Digital |
|-------------|--------------------|------------------|
| A | 135 | 5 : 4 |
| B | 140 | 9 : 5 |
| C | 180 | 4 : 5 |
| D | 160 | 13 : 19 |
| E | 150 | 7 : 8 |

Question

The table below shows the total number of watches (Analog + Digital) sold by five shopkeepers on Monday along with the ratio of analog to digital watches sold by each. Shopkeeper D sells only two brands of watches: Fossil and Titan. If $ $\frac{5}{13} $$ of the analog watches and $ $\frac{6}{19} $$ of the digital watches sold by D are Fossil brand, find the total number of Titan watches sold by shopkeeper D on Monday.

| Shopkeepers | Total Sold Watches | Analog : Digital |
|-------------|--------------------|------------------|
| A           | 135                | 5 : 4            |
| B           | 140                | 9 : 5            |
| C           | 180                | 4 : 5            |
| D           | 160                | 13 : 19          |
| E           | 150                | 7 : 8            |

Accepted Answer

**Solution:**

1. Total watches sold by shopkeeper D = 160.

2. The ratio of analog to digital watches sold by D is 13:19.

3. Let the number of analog watches sold be $13x$ and digital watches be $19x$.

4. Total watches sold: $13x + 19x = 32x = 160$.

5. Solving for $x$:
   $$
  $ 32x = 160 \implies x = \frac{160}{32} = 5$
   $$

6. Number of analog watches = $$13 \times 5 = 65$$.

7. Number of digital watches = $$19 \times 5 = 95$$.

8. Fossil analog watches = $$\frac{5}{13} \times 65 = 5 \times 5 = 25$$.

9. Fossil digital watches = $$\frac{6}{19} \times 95 = 6 \times 5 = 30$$.

10. Total Fossil watches = $25 + 30 = 55$.

11. Total Titan watches = Total watches - Fossil watches = $160 - 55 = 105$.

**Answer:** 105

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