Add or subtract the following without using a calculator. Crien mothod of estimation: \( \begin{array}{lll}\text { d) } 98+17 & \text { b) } 154+289 & \text { c) } 890+855 \\ \text { d) } 176+239 & \text { e) } 482-111 & \text { f) } 597-249 \\ \text { g) } 687-190 & \text { h) } 284-179 & \text { i) } 6248-4555\end{array} \) Iuttiply or divide the following without using a calculator. Check your answers using estimation: \( \begin{array}{lll}\text { a) } 65 \times 7 & \text { b) } 145 \times 60 & \text { c) } 800 \times 300 \\ \text { j) } 73 \times 8 & \text { e) } 423 \div 9 & \text { f) } 184 \times 6 \\ \text { 1) } 720 \div 6 & \text { b) } 159 \div 3 & \text { (i) } 984 \div 4\end{array} \)

Sure! Let's tackle each problem step-by-step using estimation methods to verify our calculations. We'll start with addition and subtraction, followed by multiplication and division. --- ## **Addition and Subtraction** ### **d) 98 + 17** **Exact Calculation:** \[ 98 + 17 = 115 \] **Estimation Method:** - **Round numbers:** Round 98 to 100 and 17 to 20. - **Add estimated numbers:** \( 100 + 20 = 120 \). - **Check:** The exact answer (115) is close to the estimate (120). ### **b) 154 + 289** **Exact Calculation:** \[ 154 + 289 = 443 \] **Estimation Method:** - **Round numbers:** Round 154 to 150 and 289 to 290. - **Add estimated numbers:** \( 150 + 290 = 440 \). - **Check:** The exact answer (443) is very close to the estimate (440). ### **c) 890 + 855** **Exact Calculation:** \[ 890 + 855 = 1,745 \] **Estimation Method:** - **Round numbers:** Round 890 to 900 and 855 to 850. - **Add estimated numbers:** \( 900 + 850 = 1,750 \). - **Check:** The exact answer (1,745) is very close to the estimate (1,750). ### **d) 176 + 239** **Exact Calculation:** \[ 176 + 239 = 415 \] **Estimation Method:** - **Round numbers:** Round 176 to 180 and 239 to 240. - **Add estimated numbers:** \( 180 + 240 = 420 \). - **Check:** The exact answer (415) is very close to the estimate (420). ### **e) 482 - 111** **Exact Calculation:** \[ 482 - 111 = 371 \] **Estimation Method:** - **Round numbers:** Round 482 to 480 and 111 to 110. - **Subtract estimated numbers:** \( 480 - 110 = 370 \). - **Check:** The exact answer (371) is very close to the estimate (370). ### **f) 597 - 249** **Exact Calculation:** \[ 597 - 249 = 348 \] **Estimation Method:** - **Round numbers:** Round 597 to 600 and 249 to 250. - **Subtract estimated numbers:** \( 600 - 250 = 350 \). - **Check:** The exact answer (348) is very close to the estimate (350). ### **g) 687 - 190** **Exact Calculation:** \[ 687 - 190 = 497 \] **Estimation Method:** - **Round numbers:** Round 687 to 690 and 190 to 190. - **Subtract estimated numbers:** \( 690 - 190 = 500 \). - **Check:** The exact answer (497) is very close to the estimate (500). ### **h) 284 - 179** **Exact Calculation:** \[ 284 - 179 = 105 \] **Estimation Method:** - **Round numbers:** Round 284 to 280 and 179 to 180. - **Subtract estimated numbers:** \( 280 - 180 = 100 \). - **Check:** The exact answer (105) is very close to the estimate (100). ### **i) 6,248 - 4,555** **Exact Calculation:** \[ 6,248 - 4,555 = 1,693 \] **Estimation Method:** - **Round numbers:** Round 6,248 to 6,200 and 4,555 to 4,600. - **Subtract estimated numbers:** \( 6,200 - 4,600 = 1,600 \). - **Check:** The exact answer (1,693) is close to the estimate (1,600). --- ## **Multiplication and Division** ### **a) 65 × 7** **Exact Calculation:** \[ 65 \times 7 = 455 \] **Estimation Method:** - **Round numbers:** Round 65 to 60. - **Multiply estimated numbers:** \( 60 \times 7 = 420 \). - **Check:** The exact answer (455) is close to the estimate (420). ### **b) 145 × 60** **Exact Calculation:** \[ 145 \times 60 = 8,700 \] **Estimation Method:** - **Round numbers:** Round 145 to 150 and 60 to 60. - **Multiply estimated numbers:** \( 150 \times 60 = 9,000 \). - **Check:** The exact answer (8,700) is close to the estimate (9,000). ### **c) 800 × 300** **Exact Calculation:** \[ 800 \times 300 = 240,000 \] **Estimation Method:** - **Round numbers:** Already multiples of 100; no need to round. - **Multiply estimated numbers:** \( 800 \times 300 = 240,000 \). - **Check:** The estimate matches the exact answer. ### **j) 73 × 8** **Exact Calculation:** \[ 73 \times 8 = 584 \] **Estimation Method:** - **Round numbers:** Round 73 to 70. - **Multiply estimated numbers:** \( 70 \times 8 = 560 \). - **Check:** The exact answer (584) is close to the estimate (560). ### **e) 423 ÷ 9** **Exact Calculation:** \[ 423 \div 9 = 47 \text{ (since } 9 \times 47 = 423\text{)} \] **Estimation Method:** - **Round numbers:** Round 423 to 420. - **Divide estimated numbers:** \( 420 \div 10 = 42 \) (using 10 for easier calculation, knowing it's close to 9). - **Adjust:** Since we divided by a slightly larger number (10 instead of 9), the actual answer should be slightly higher. - **Check:** The exact answer (47) is slightly higher than the estimate (42), which aligns with our expectation. ### **f) 184 × 6** **Exact Calculation:** \[ 184 \times 6 = 1,104 \] **Estimation Method:** - **Round numbers:** Round 184 to 180. - **Multiply estimated numbers:** \( 180 \times 6 = 1,080 \). - **Check:** The exact answer (1,104) is close to the estimate (1,080). ### **1) 720 ÷ 6** **Exact Calculation:** \[ 720 \div 6 = 120 \] **Estimation Method:** - **Round numbers:** Already divisible by 6; no need to round. - **Divide estimated numbers:** \( 720 \div 6 = 120 \). - **Check:** The estimate matches the exact answer. ### **b) 159 ÷ 3** **Exact Calculation:** \[ 159 \div 3 = 53 \] **Estimation Method:** - **Round numbers:** Round 159 to 160. - **Divide estimated numbers:** \( 160 \div 3 \approx 53.3 \). - **Check:** The exact answer (53) is very close to the estimate (~53.3). ### **i) 984 ÷ 4** **Exact Calculation:** \[ 984 \div 4 = 246 \] **Estimation Method:** - **Round numbers:** Round 984 to 1,000. - **Divide estimated numbers:** \( 1,000 \div 4 = 250 \). - **Check:** The exact answer (246) is very close to the estimate (250). --- ## **Summary** By using estimation methods such as rounding to the nearest ten, hundred, or thousand, we can quickly verify that our exact calculations are reasonable. This approach is especially useful for checking work without a calculator. If you have any further questions or need additional explanations, feel free to ask!