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For the ISEE and other standardized tests with math sections, estimation is your friend. Estimation and rounding will reduce the amount of time you spend making calculations, which allows more time to complete difficult questions. Rounding a number up or down will make it easier to calculate, but it is important that the estimated number be as close in value to the original number as possible. In general, you choose a specific place value to estimate, using the next lowest place value (to the right of it) as your guide. If you want to estimate a number at the second place value, then the third place value is your guide; if you’re estimating the third place value, then you use the fourth place value for your guide and so on. There are two simple rules for rounding a specific place value: If the next lowest place value is 5 or greater, you round that place value up. If the next lowest place value is 4 or lower, you round that place value down. Rounding Whole Numbers vs Decimals The process for rounding decimals and rounding whole numbers is slightly different. For rounding whole numbers, you take the place value you want to estimate and use the next lowest place value. If the next lowest place value is 5 or greater, add one to the estimated place value and change all digits to right to 0. If the next lowest place value is less than 5, do not change the estimated place value, but still change all digits to the right to 0. For decimals, you take the place value you want to estimate and use the next lowest place value. If the next lowest place value is 5 or greater, add one to the estimated place value and discard all digits to the right. If the next lowest place value is less than 5, do not change the estimated place value, but still drop all digits to the right. Rounding Examples For example, let’s say you are asked to find the product of 3,984,342,234,456,678 and 415, and you have 4 possible choices for the answer. Without a calculator, this will take you ages to multiply. However, if you can estimate these values, you might find the problem easier to crack: When you want to estimate a number closer to the original number’s value, look at the three highest place values. Think about how much your estimation will change the original number: If the estimation of the highest place value results in a very different number, look to the next highest place value. Generally, estimating the second highest place value will result in a number that is truer to the essence of the original. As we can see in the above examples, rounding to the first place value gives us a less precise estimate, which may alter your answer to a significant degree. Other Estimating Examples Estimation can also help you solve different sorts of problems. Say you have to find where the function f(x) = x² + 7x + 10 crosses the x-axis. To do this, you need to find the values of x when f(x) is 0. Suppose the multiple choice answers are: a) x = 5, 2 b) x = 15, -12 c) x = -5, -2 d) x = -5, 20 To solve this problem, we can use factoring. Before we do this, however, we can use estimation to know which answers are not viable. When factoring, we look for factors of c, which in this case is 10. Since we know that the factors of 10 will not be larger than 10, we can cross out answers (B) and (D) immediately. Now we only have to choose between (A) and (C). Since the factors also need to add up to 7 (since that is the coefficient of x!), we know the factors should be positive, thus (A) is the correct answer. Estimation in Geometry We can also use estimation in geometry. For example, you may need to estimate the length of a side of a triangle. In the above triangle, we can estimate the length of the third side using our knowledge of geometric principles: We know that the third side of the triangle must be smaller than the other two sides combined, so it must be less than 6.5. Additionally, given that the two known sides form an obtuse angle, we know that the third side must be larger than the base length of 4. Thus, we can estimate that the third length, x, is: 4 < x < 6.5. Practice Problems Round the following decimals the nearest hundredth. 0.01767 0.35512 0.8334 Round the following numbers to the 2nd highest place value. 2,859 99,481 853,402 Round 78,746,910 to the 1st, 2nd, and 3rd place value. Which estimation seems best to use? What is the product of 625,424,913,788 and 8,956? a) 5. 6 × 1014 b) 5.6 × 1015 c) 5.6 × 1016 d) 5.6 × 1017 Which of the following is NOT a valid estimation of 72.4565? a) 72.45 b) 70 c) 72.5 d) 72.457 Answers to Practice Problems Question 1 0.02 0.36 0.83 Question 2 2,900 99,000 850,000 Question 3 1st place value = 80,000,000 2nd place value = 79,000,000 3rd place value = 78,700,000 The 2nd place value seems like the best estimation to use, which is often the case. This is true because it alters the value the least, while maintaining more information. B A
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