Sum Products Ea5F25

1. **State the problem:** Calculate the value of the sum $$0.05\times16 + 0.06\times15 + 0.07\times15 + 0.08\times16 + 0.09\times15 + 0.10\times15 + 0.11\times15 + 0.12\times16 + 0.13\times15 + 0.14\times15 + 0.15\times15 + 0.16\times16 + 0.17\times15 + 0.18\times15 + 0.19\times15 + 0.20\times16 + 0.21\times15 + 0.22\times15 + 0.23\times15 + 0.24\times16 + 0.25\times15 + 0.26\times15 + 0.27\times16 + 0.28\times15 + 0.29\times15 + 0.30\times15 + 0.31\times16 + 0.32\times15 + 0.33\times15 + 0.34\times15 + 0.35\times16 + 0.36\times15 + 0.37\times15 + 0.38\times15 + 0.39\times15 + 0.40\times16 + 0.41\times15 + 0.42\times15 + 0.43\times16 + 0.44\times15 + 0.45\times15 + 0.46\times15 + 0.47\times15 + 0.48\times15 + 0.49\times15 + 0.50\times16 + 0.51\times15 + 0.52\times15 + 0.53\times15$$ 2. **Approach:** Multiply each decimal by its corresponding integer and then sum all the products. 3. **Calculate each product:** - $0.05 \times 16 = 0.80$ - $0.06 \times 15 = 0.90$ - $0.07 \times 15 = 1.05$ - $0.08 \times 16 = 1.28$ - $0.09 \times 15 = 1.35$ - $0.10 \times 15 = 1.50$ - $0.11 \times 15 = 1.65$ - $0.12 \times 16 = 1.92$ - $0.13 \times 15 = 1.95$ - $0.14 \times 15 = 2.10$ - $0.15 \times 15 = 2.25$ - $0.16 \times 16 = 2.56$ - $0.17 \times 15 = 2.55$ - $0.18 \times 15 = 2.70$ - $0.19 \times 15 = 2.85$ - $0.20 \times 16 = 3.20$ - $0.21 \times 15 = 3.15$ - $0.22 \times 15 = 3.30$ - $0.23 \times 15 = 3.45$ - $0.24 \times 16 = 3.84$ - $0.25 \times 15 = 3.75$ - $0.26 \times 15 = 3.90$ - $0.27 \times 16 = 4.32$ - $0.28 \times 15 = 4.20$ - $0.29 \times 15 = 4.35$ - $0.30 \times 15 = 4.50$ - $0.31 \times 16 = 4.96$ - $0.32 \times 15 = 4.80$ - $0.33 \times 15 = 4.95$ - $0.34 \times 15 = 5.10$ - $0.35 \times 16 = 5.60$ - $0.36 \times 15 = 5.40$ - $0.37 \times 15 = 5.55$ - $0.38 \times 15 = 5.70$ - $0.39 \times 15 = 5.85$ - $0.40 \times 16 = 6.40$ - $0.41 \times 15 = 6.15$ - $0.42 \times 15 = 6.30$ - $0.43 \times 16 = 6.88$ - $0.44 \times 15 = 6.60$ - $0.45 \times 15 = 6.75$ - $0.46 \times 15 = 6.90$ - $0.47 \times 15 = 7.05$ - $0.48 \times 15 = 7.20$ - $0.49 \times 15 = 7.35$ - $0.50 \times 16 = 8.00$ - $0.51 \times 15 = 7.65$ - $0.52 \times 15 = 7.80$ - $0.53 \times 15 = 7.95$ 4. **Sum all products:** $$\begin{aligned} &0.80 + 0.90 + 1.05 + 1.28 + 1.35 + 1.50 + 1.65 + 1.92 + 1.95 + 2.10 + 2.25 + 2.56 + 2.55 + 2.70 + 2.85 + 3.20 + 3.15 + 3.30 + 3.45 + 3.84 + 3.75 + 3.90 + 4.32 + 4.20 + 4.35 + 4.50 + 4.96 + 4.80 + 4.95 + 5.10 + 5.60 + 5.40 + 5.55 + 5.70 + 5.85 + 6.40 + 6.15 + 6.30 + 6.88 + 6.60 + 6.75 + 6.90 + 7.05 + 7.20 + 7.35 + 8.00 + 7.65 + 7.80 + 7.95 \\ &= 206.88 \end{aligned}$$ 5. **Final answer:** The sum of the given expression is **206.88**.