Section 1.5

1. Section 1.5 Rounding. Solving Equations

2. Rounding Quite often in real-life situations, like shopping, estimating and many others, we are not really concerned with the exact answer, but we need only the ball park in which the answer falls. In these situations we use rounding to do our calculations instead of using exact numbers.

3. Example: Consider shopping at the supermarket. You pick up the following merchandise: • Gallon of milk $3.99 • One dozen of eggs $1.79 • Loaf of bread $2.09 • 2 lbs of apples $0.89 per lb • Yogurt $2.29 How much money do you need to pay for the purchase? If you have $20, would it be enough to pay?

4. Example (continued): If you encounter this problem in the store and you don’t have a calculator, then you would like to make a rough estimate and round all the numbers to closest dollar amounts. For instance, $3.99 is approximately $4, $1.79 is approximately $2, S2.09 is close to $2 and so on… So instead of adding the exact amounts we add their rounded estimates and get Actual amount (before tax) is So our rough estimate is quite close to the actual amount.

6. Solving Equations and Problems • Example 1: If you gave the waiter in a restaurant some money to pay $43 bill and got $12 change, how much money did you give? If you add the amount you paid to the amount you were left with – it will give you the original amount. • Example 2: If you had 9 feet long wooden plank, a piece of it broke and fall off, and you are left with 4 feet long plank, how long was the broken piece? If you add the length of the broken piece to the length of the piece left you will get the length of the original plank. Or if you subtract from the original length of 9 feet the length of the leftover piece, you will get the length of the broken piece.

7. Variables and Equations • In mathematics we denote the unknown quantity with the variable – usually a letter x, y, z, w, t and write an equation describing the situation, using the known quantities and variables instead of the unknown. • For instance the first example could be described by the following equation: If xrepresents the original unknown amount, then • The second example could be described by the following equation: where l represents an unknown length of the broken piece.

8. Equations Example: Example: Number 4 is a solution of an equation since when we replace x with 4 we shall get a true statement

9. Example: Example: Consider the simple equation Solution of this equation is obvious, it is number 5, since if we replace the variable x with 5, the equation will turn into a true statement. true The other numbers, like 3, 0, -7, 14 etc., are not the solutions, since if we replace x with these values, an equation will not turn into a true statement: false; false; false; false;

10. Equation as Balanced Scales We can always use a powerful visual representation of an equation; we can visualize this equation as a balanced pair of scales with x and 5 measured in kilograms.

11. Equation as Balanced Scales If we add 3 kg to the scale on the left-hand side, the scales will balance as long as we add 3 kg to the scale on the right-hand side.  That is, x + 3 = 8.

12. Equation as Balanced Scales Also, if we subtract the same weight, say 3 kg, from each side of the balance, the scales will remain balanced.  That is, x – 3 = 2.

13. Equation as Balanced Scales If we double the weight in the scale on the left-hand side, the scales will balance as long as we double the weight in the scale on the right-hand side.  That is, 2x = 10.

14. Equation as Balanced Scales Also, if we halve the weight in each scale of the balance, the scales will remain balanced.     That is, It is not hard to check out that all of the resulting equations have the same solution 5.

15. Operations in Solving Equations Notice that an equation like x = 5is easy to solve (find a solution). The solution is just a number on the right hand side of the equation. So if we can rewrite every equation in such a form without changing a solution, we can easily read the solution of the resulting equation. To write an equation in such a form we will use allowed operations.

16. Example: Solve an equation: If we add 7 to both sides of an equation we shall get Check the solution: true

17. Example: Solve an equation: If we divide both sides of an equation by 3 we shall get Check the solution: true

18. Example: Solve an equation: first we want to leave the term with xalone on one side of an equation; so we add 5 to both sides of an equation and get now we divide both sides of an equation by 2 and get Check the solution: true

19. Solving an Equation