Pharmacy+Calculations

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 Utilize the following examples to expand your knowledge

Dividing Whole Numbers, with Remainders Example: 1400 ÷ 7.. Since 14 ÷ 7 = 2, and 1400 is 100 times greater than 14, the answer is 2 × 100 = 200. Many problems are similar to the above example, where the answer is easily obtained by adding on or taking off an appropriate number of 0's. Others are more complicated. Example: 4934 ÷ 6. Use long division.

So the answer is 822 with a remainder of 2, written 822 R2. To double-check that the answer is correct, multiply the quotient by the divisor and add the remainder: (822 × 6) + 2 = 4932 + 2 = 4934.


 * //__ Problem 16 __//**

To divide by a decimal, multiply that decimal by a power of 10 great enough to obtain a whole number. Multiply the dividend by that same power of 10. Then the problem becomes one involving division by a whole number instead of division by a decimal. Example: 0.144 ÷ 0.12 Multiplying the divisor (0.12) and the dividend (0.144) by 100, then dividing, gives the same result.

The answer is 1.2. Be aware that some problems are less difficult and do not require this procedure.

 Decimal numbers such as 3.762 are used in situations which call for more precision than whole numbers provide. As with whole numbers, a digit in a decimal number has a value which depends on the place of the digit. The places to the left of the decimal point are ones, tens, hundreds, and so on, just as with whole numbers. This table shows the decimal place value for various positions: Note that adding extra zeros to the right of the last decimal digit does not change the value of the decimal number.
 * //__ Problem 17 __//**
 * **__ Place (underlined) __** || **__ Name of Position __** ||
 * __ 1 __ .234567 || Ones (units) position ||
 * 1.__2__34567 || Tenths ||
 * 1.2__3__4567 || Hundredths ||
 * 1.23__4__567 || Thousandths ||
 * 1.234__5__67 || Ten thousandths ||
 * 1.2345__6__7 || Hundred Thousandths ||
 * 1.23456__7__ || Millionths ||

 To subtract decimals, line up the decimal points and then follow the rules for adding or subtracting whole numbers, placing the decimal point in the same column as above. When one number has more decimal places than another, use 0's to give them the same number of decimal places. Example: 18.2 - 6.008 1) Line up the decimal points. - 6.008 || 2) Add extra 0's, using the fact that 18.2 = 18.200 - 6.008 || 3) Subtract. - __6.008__ 12.192 ||
 * //__ Problem 18 __//**
 * 18.2
 * 18.200
 * 18.200


 * //__ Problems 19&20 __//**

Rounding Decimal Numbers
To round a number to any decimal place value, we want to find the number with zeros in all of the lower places that is closest in value to the original number. As with whole numbers, we look at the digit to the right of the place we wish to round to. Note: When the digit 5, 6, 7, 8, or 9 appears in the ones place, round up; when the digit 0, 1, 2, 3, or 4 appears in the ones place, round down. Examples: Rounding 1.19 to the nearest tenth gives 1.2 (1.20). Rounding 1.545 to the nearest hundredth gives 1.55. Rounding 0.1024 to the nearest thousandth gives 0.102. Rounding 1.80 to the nearest one gives 2. Rounding 150.090 to the nearest hundred gives 200. Rounding 4499 to the nearest thousand gives 4000.

**// Converting Improper Fractions to Mixed Numbers //** To change an improper fraction into a mixed number, divide the numerator by the denominator. The remainder is the numerator of the fractional part. Examples: 11/4 = 11 ÷ 4 = 2 //r//3 = 2 3/4 13/2 = 13 ÷ 2 = 6 //r//1 = 6 1/2 **// Writing a Fraction as a Decimal //** Method 1 - Convert to an equivalent fraction whose denominator is a power of 10, such as 10, 100, 1000, 10000, and so on, then write in decimal form. Examples: 1/4 = (1 × 25)/(4 × 25) = 25/100 = 0.25   3/20 = (3 × 5)/(20 × 5) = 15/100 = 0.15    9/8 = (9 × 125)/(8 × 125) = 1125/1000 = 1.125    Method 2 - Divide the numerator by the denominator. Round to the decimal place asked for, if necessary. Example: 13/4 = 13 ÷ 4 = 3.25   Example: Convert 3/7 to a decimal. Round to the nearest thousandth. We divide one decimal place past the place we need to round to, then round the result. 3/7 = 3 ÷ 7 = 0.4285…   which equals 0.429 when rounded to the nearest thousandth. Example: Convert 4/9 to a decimal. Round to the nearest hundredth. We divide one decimal place past the place we need to round to, then round the result. 4/9 = 4 ÷ 9 = 0.4444…   which equals 0.44 when rounded to the nearest hundredth. **// Rounding a Fraction to the Nearest Hundredth //** Divide to the thousandths place. If the last digit is less than 5, drop it. This is particularly useful for converting a fraction to a percent, if we want to convert to the nearest percent. 1/3 = 1 ÷ 3 = 0.333… which rounds to 0.33 If the last digit is 5 or greater, drop it and round up. 2/7 = 2 ÷ 7 = 0.285 which rounds to 0.29 **// Adding and Subtracting Fractions //** If the fractions have the same denominator, their sum is the sum of the numerators over the denominator. If the fractions have the same denominator, their difference is the difference of the numerators over the denominator. We do not add or subtract the denominators! Reduce if necessary. Examples: 3/8 + 2/8 = 5/8   9/2 - 5/2 = 4/2 = 2    If the fractions have different denominators: 1) First, find the least common denominator. 2) Then write equivalent fractions using this denominator. 3) Add or subtract the fractions. Reduce if necessary.   Example:    3/4 + 1/6 = ?    The least common denominator is 12.    3/4 + 1/6 = 9/12 + 2/12 = 11/12.    Example:    9/10 - 1/2 = ?    The least common denominator is 10.    9/10 - 1/2 = 9/10 - 5/10 = 4/10 = 2/5.    Example:    2/3 + 2/7 = ?    The least common denominator is 21    2/3 + 2/7 = 14/21 + 6/21 = 20/21.   **// Adding and Subtracting Mixed Numbers //**   To add or subtract mixed numbers, simply convert the mixed numbers into improper fractions, then add or subtract them as fractions.    Example:    9 1/2 + 5 3/4 = ?    Converting each number to an improper fraction, we have 9 1/2 = 19/2 and 5 3/4 = 23/4.    We want to calculate 19/2 + 23/4. The LCM of 2 and 4 is 4, so    19/2 + 23/4 = 38/4 + 23/4 = (38 + 23)/4 = 61/4.    Converting back to a mixed number, we have 61/4 = 15 1/4.    The strategy of converting numbers into fractions when adding or subtracting is often useful, even in situations where one of the numbers is whole or a fraction. Example: 13 - 1 1/3 = ?   In this situation, we may regard 13 as a mixed number without a fractional part. To convert it into a fraction, we look at the denominator of the fraction 4/3, which is 1 1/3 expressed as an improper fraction. The denominator is 3, and 13 = 39/3. So 13 - 1 1/3 = 39/3 - 4/3 = (39-4)/3 = 35/3, and 35/3 = 11 2/3. Example: 5 1/8 - 2/3 = ?   This time, we may regard 2/3 as a mixed number with 0 as its whole part. Converting the first mixed number to an improper fraction, we have 5 1/8 = 41/8. The problem becomes 5 1/8 - 2/3 = 41/8 - 2/3 = 123/24 - 16/24 = (123 - 16)/24 = 107/24.   Converting back to a mixed number, we have 107/24 = 4 11/24. Example: 92 + 4/5 = ?   This is easy. To express this as a mixed number, just put the whole number and the fraction side by side. The answer is 92 4/5. **// Multiplying Fractions and Whole Numbers //** To multiply a fraction by a whole number, write the whole number as an improper fraction with a denominator of 1, then multiply as fractions. Example: 8 × 5/21 = ?   We can write the number 8 as 8/1. Now we multiply the fractions. 8 × 5/21 = 8/1 × 5/21 = (8 × 5)/(1 × 21) = 40/21   Example: 2/15 × 10 = ?   We can write the number 10 as 10/1. Now we multiply the fractions. 2/15 × 10 = 2/15 × 10/1 = (2 × 10)/(15 × 1) = 20/15 = 4/3  **// Multiplying Fractions and Fractions //** When two fractions are multiplied, the result is a fraction with a numerator that is the product of the fractions' numerators and a denominator that is the product of the fractions' denominators. Example: 4/7 × 5/11 = ?   The numerator will be the product of the numerators: 4 × 5, and the denominator will be the product of the denominators: 7 × 11. The answer is (4 × 5)/(7 × 11) = 20/77. Remember that like numbers in the numerator and denominator cancel out. Example: 14/15 × 15/17 = ?   Since the 15's in the numerator and denominator cancel, the answer is    14/15 × 15/17 = 14/1 × 1/17 = (14 × 1)/(1 × 17) = 14/17 Example: 4/11 × 22/36 = ?   In the solution below, first we cancel the common factor of 11 in the top and bottom of the product, then we cancel the common factor of 4 in the top and bottom of the product. 4/11 × 22/36 = 4/1 × 2/36 = 1/1 × 2/9 = 2/9
 * //__ Problems 21-25 __//**

**// Multiplying Mixed Numbers //** To multiply mixed numbers, convert them to improper fractions and multiply. Example: 4 1/5 × 2 2/3 = ?.   Converting to improper fractions, we get 4 1/5 = 21/5 and 2 2/3 = 8/3. So the answer is   4 1/5 × 2 2/3 = 21/5 × 8/3 = (21 × 8)/(5 × 3) = 168/15 = 11 3/15. Examples: 3/4 × 1 1/8 = 3/4 × 9/8 = 27/32.   3 × 7 3/4 = 3 × 31/4 = (3 × 31)/4 = 93/4 = 23 1/4.   **// Reciprocal //** The reciprocal of a fraction is obtained by switching its numerator and denominator. To find the reciprocal of a mixed number, first convert the mixed number to an improper fraction, then switch the numerator and denominator of the improper fraction. Notice that when you multiply a fraction and its reciprocal, the product is always 1. Example: Find the reciprocal of 31/75. We switch the numerator and denominator to find the reciprocal: 75/31. Example: Find the reciprocal of 12 1/2. First, convert the mixed number to an improper fraction: 12 1/2 = 25/2. Next, we switch the numerator and denominator to find the reciprocal: 2/25. **// Dividing Fractions //** To divide a number by a fraction, multiply the number by the reciprocal of the fraction. Examples: 7 ÷ 1/5 = 7 × 5/1 = 7 × 5 = 35   1/5 ÷ 16 = 1/5 ÷ 16/1 = 1/5 × 1/16 = (1 × 1)/(5 × 16) = 1/80    3/5 ÷ 7/12 = 3/5 × 12/7 = (3 × 12)/(5 × 7) = 36/35 or 1 1/35

**// Dividing Mixed Numbers //** To divide mixed numbers, you should always convert to improper fractions, then multiply the first number by the reciprocal of the second. Examples: 1 1/2 ÷ 3 1/8 = 3/2 ÷ 25/8 = 3/2 × 8/25 = (3 × 8)/(2 × 25) = 24/50   1 ÷ 3 3/5 = 1/1 ÷ 18/5 = 1/1 × 5/18 = (1 × 5)/(1 × 18) = 5/18    3 1/8 ÷ 2 = 25/8 ÷ 2/1 = 25/8 × 1/2 = (25 × 1)/(8 × 2) = 25/16 or 1 9/16. **// Repeating Decimals //** Every fraction can be written as a decimal. For example, 1/3 is 1 divided by 3. If you use a calculator to find 1 ÷ 3, the calculator returns 0.333333... This is called a repeating decimal. To represent the idea that the 3's repeat forever, one uses a horizontal bar (overstrike) as shown below:

Example: What is the repeating decimal for 1/7 ? Dividing 7 into 1, we get 0.142857142..., and we see the pattern begin to repeat with the second 1, so.

**// Problem 28-52 //** A percent is a ratio of a number to 100. A percent can be expressed using the percent symbol %. Example: 10 percent or 10% are both the same, and stand for the ratio 10:100. **// Percent as a fraction //** A percent is equivalent to a fraction with denominator 100. Example: 5% of something = 5/100 of that thing. Example: 2 1/2% is equal to what fraction? Answer: 2 1/2% = (2 1/2)/100 = 5/200 = 1/40   Example: 52% most nearly equals which one of 1/2, 1/4, 2, 8, or 1/5? Answer: 52% = 52/100. This is very close to 50/100, or 1/2. Example: 13/25 is what %? We want to convert 13/25 to a fraction with 100 in the denominator: 13/25 = (13 × 4)/(25 × 4) = 52/100, so 13/25 = 52%. Alternatively, we could say: Let 13/25 be //n//%, and let us find //n//. Then 13/25 = //n///100, so cross multiplying, 13 × 100 = 25 × //n//, so 25//n// = 13 × 100 = 1300. Then 25//n// ÷ 25 = 1300 ÷ 25, so //n// = 1300 ÷ 25 = 52. So 13/25 = //n//% = 52%. Example: 8/200 is what %? Method 1: 8/200 = (4 × 2)/(100 × 2), so 8/200 = 4/100 = 4%. Method 2: Let 8/200 be n%. Then 8/200 = //n///100, so 200 × //n// = 800, and 200//n// ÷ 200 = 800 ÷ 200 = 4, so //n//% = 4%. Example: Write 80% as a fraction in lowest terms. 80% = 80/100, which is equal to 4/5 in lowest terms. **// Percent as a decimal //** Percent and hundredths are basically equivalent. This makes conversion between percent and decimals very easy. To convert from a decimal to a percent, just move the decimal 2 places to the right. For example, 0.15 = 15 hundredths = 15%. Example: 0.0006 = 0.06%   Converting from percent to decimal form is similar, only you move the decimal point 2 places to the left. You must also be sure, before doing this, that the percentage itself is expressed in decimal form, without fractions. Example: Express 3% in decimal form. Moving the decimal 2 to the left (and adding in 0's to the left of the 3 as place holders,) we get 0.03. Example: Express 97 1/4% in decimal form. First we write 97 1/4 in decimal form: 97.25. Then we move the decimal 2 places to the left to get 0.9725, so 97 1/4% = 0.9725. This makes sense, since 97 1/4% is nearly 100%, and 0.9725 is nearly 1. Example: What is 5/8 as a percent? Get your calculator and type in "5 / 8 =", the calculator should show 0.625, then multiply by 100 and your answer is: 62.5% (remember to put the "%" so people know it is "per 100") Of course you can do the division in your head or on paper if you don't have a calculator! Another Method Because percent means "per 100", you can try to convert the fraction to **?**/**100** form. Follow these steps:
 * // What is a Percent? //**
 * Step 1: Find a number you can multiply by the bottom of the fraction to get 100. ||


 * Step 2: Multiply both top and bottom of the fraction by that number. ||


 * Step 3. Then write down just the top number with the "%" sign. ||

Example 1: Express 3/4 as a Percent Step 1: We can multiply 4 by 25 to become 100 Step 2: Multiply top and bottom by 25:
 * ×25 ||




 * 3 ||  || = ||   || 75 ||




 * 4 ||  || 100 ||




 * ×25 ||

Step 3: Write down 75 with the percent sign: Answer = 75% Example 2: Express 3/16 as a Percent Step 1: We have to multiply 16 by 6.25  to become 100 Step 2: Multiply top and bottom by 6.25:
 * ×6.25 ||




 * 3 ||  || = ||   || 18.75 ||




 * 16 ||  || 100 ||




 * ×6.25 ||

Step 3: Write down 18.75 with the percentage sign: Answer = 18.75% To change a Percent to a Decimal Move the decimal point two places to the left. Examples: 75% converts to 0.75 40% converts to 0.4 230% converts to 2.3 8% converts to 0.08 3.1% converts to 0.031 1.2% converts to 0.012

**//__ To turn a Decimal to a Percent __//** Move the decimal point two places to the right. Examples: 0.32 becomes 32% 0.07 becomes 7% 0.6 becomes 60% 1.25 becomes 125% 0.083 becomes 8.3%

**//__ To change a Fraction into a Decimal __//** On a calculator, enter the numerator, hit the divide key, enter the denominator, and hit equals. If doing the problem longhand, put the numerator under the division box and the denominator on the outside. For more information on doing longhand division, check the wikiHow article, [|How to Change a Common Fraction Into a Decimal].

**//__ To change a Decimal into a Fraction __//** Take the decimal, drop the decimal point, and place the result into the numerator (top number) of a fraction. To determine the denominator (bottom number), write a 1, followed by zeros --- as many zeroes as it takes to match the original length of the decimal. Examples: 0.75 becomes 75/100 0.034 becomes 34/1000 2.5 becomes 25/10 0.6 becomes 6/10 0.0006 becomes 6/10000 1.25 becomes 125/100 Take your fraction down to lowest terms, if necessary. For example, 75/100 simplifies to 3/4.

**//__ To change a Fraction into Percent __//** Divide the top of the fraction by the bottom. You should now have a decimal. Move the decimal point two places to the right (or multiply by 100) to make it into a percentage. Example: 5/16 becomes 5 ÷ 16 which equals 0.3125 which turns into 31.25%

**//__ Percent to a Fraction __//** Example:36% turns to 36/100 then simplify. The answer would be 9/25.

1. 498 + 254 = A: 732 B: 752 C: 742 D: 753

2. 5997 + 4327 = A: 10,234 B: 10,442 C: 10,324 D: 10,554

3. 3,788 + 2,265 = A: 6,053 B: 5,053 C: 6,054 D: 6,044

4. 7,769 – 3,628 = A: 4,133 B: 4,142 C: 4,041 D: 4,141

5. 783 – 396 = A: 387 B: 377 C: 397 D: 407

6. 3,077 – 2,488 = A: 579 B: 590 C: 578 D: 589

7. 198 x 6 = A: 1,288 B: 1,268 C: 1,378 D: 1,188

8. 519 x 19 = A: 8,981 B: 9,861 C: 10,681 D: 9,691

9. 301 x 35 = A: 10,535 B: 8,544 C: 10,574 D: 10,775

10. 469 ÷ 3 = A: 151 r2 B: 146 r1 C: 157 r2 D: 156 r1

11. 303 ÷ 4 = A: 73 r3 B: 75 r3 C: 64 r1 D: 72 r3

12. 14,883 ÷ 36 = A: 410 r12 B: 412 r14 C: 413 r15 D: 415 r17

13. Express 546/1000 as a decimal. A: .0546 B: .546 C: 54.6 D: 546.000

14. .89 + 3.8 + .036 = A: 47.26 B: 4.726 C: 48.26 D: 4.826

15. .27 x .4 = A: 1.08 B: .0108 C: 10.8 D: .108

16. .84 ÷ .6 = A: 14 B: .14 C: .014 D: 1.4

17. Express the hundredths place in .2380 A: 3 B: 2 C: 0 D: 8

18. .37 - .19 = A: .19 B: .019 C: 17 D: .18

19. Round 5.788 to the nearest hundredth. A: 5.8 B: 5.79 C: 5.80 D: 5.77

20. Which is the equivalent decimal number for three hundred sixty-four thousandths? A: .364 B: 364,000 C: 3.6400 D: .0364

21. 3 1/3 + 4 + 2/5 = A: 8 ¾ B: 7 2/3 C: 7 ½ D: 7 11/15

22. 4 ¾ - 2 ½ = A: 2 ¼ B: 2 ½ C: 3 ¼ D: 2 1/3

23. 4 1/5 x ¾ x 6 = A: 17 4/5 B: 16 2/3 C: 20 ½ D: 18 9/10

24. 2/7 ÷ 1/9 = A: 18/7 B: 7/2 C: 7/18 D: 2/63

25. Which of the following is correct? A: 9/15 = 3/4 B: 12/15 = 4/5 C: 12/16 = 2/3 D: 3/15 = 1/4

26. Solve for x: 4/x = 1/5 A: 15 B: 18 C: 12 D: 20

27. Reduce 12/156 to lowest terms. A: 1/12 B: 2/7 C: 1/13 D: 1/10

28. Express 17/6 as a mixed fraction. A: 2 5/6 B: 3 1/6 C: 2 2/6 D: 2 1/6

29. Express seventy four hundredths as a percentage. A: 740% B: 7.4% C: 74% D: .74%

30. 15% of 120 = A: 16 B: 15 C: 18 D: 20

31. 9 is what percent of 72? A: 10% B: 15% C: 18% D: 12.5%

32. 4 is what percent of 50? A: 9% B: 8% C: 10% D: 4%

33. Four tenths of 18 equals: A: 6.8 B: 7 C: 7.2 D: 7.4

34. .6% of 18 equals: A: 1.08 B: 10.8 C: .108 D: .0108

35. The ratio of 7:4 = (?)% A: 160% B: 67% C: 175% D: 125%

36. 2/4 = (?)% x 2/5 A: 125% B: 110% C: 105% D: 135%

37. Express 4/16 as a decimal. A: .5 B: .025 C: .25 D: .05

38. Express .24 as a percentage. A: 2.4% B: .24% C: .024% D: 24%

39. Express 14/4 as a percentage. A: 30% B: 333% C: 300% D: 350%

40. Express 5/25 as a percentage. A: 20% B: 15% C: 25% D: 30%

41. 1.5/50% = ? A: 3 B: 4.5 C: 6 D: 1.5

42. Express .075 as a percentage. A: 75% B: 750% C: .75% D: 7.5%

43. Express the ratio of 6:14 as a percentage. A: 40% B: 50% C: 43% D: 55%

44. Express 3/8 as a percentage. A: 33% B: 30% C: 37.5% D: 40%

45. Express 8.6% as a reduced fraction. A: 21/250 B: 86/1,000 C: 22/84 D: 43/500

46. Express 160% as a decimal. A: 16 B: 1.6 C: .016 D: .16

47. Solve for x: 24 is 60% of x A: 4 B: 400 C: 20 D: 40

48. 8% of x is equal to 48. Solve for x. A: 600 B: 540 C: 480 D: 640

49. Express 32/96 as a reduced common fraction. A: ½ B: 2/5 C: 3/5 D: 1/3

50. Express 37% as a decimal. A: 37.0 B: .37 C: 3.7 D: .037

51. 8 is 40% of x. Solve for x. A: 50 B: 40 C: 20 D: 30

52. 25% of x is 50. Solve for x. A: 260 B: 220 C: 240 D: 200

53. 4b + 2b – 5b = A: 2b B: b C: -b D: -2b

54. (5y2 – 4y) + (2y2 + 6y -9) = A: 6y2 +2y – 9 B: 7y2 + 2y – 9 C: 6y2 + y – 9 D: 5y2 + 2y – 9

55. 4x – 6 = 42. Solve for x. A: 12 B: 10 C: 8 D: 14

56. 5(x – 6) = 4x + 4. Solve for x. A: 28 B: 32 C: 30 D: 34

57. 5ab2 + 3ab2 = A: 8a2b4 B: 8 C: 8a2b2 D: 8ab2

58. 5(3y – 5) + 2 – 6(y + 7) = A: 7y – 55 B: 9y – 65 C: 15y – 55 D: 10y – 65

59. 7x + 6 = 34. Solve for x. A: 4 B: 6 C: -2 D: 2

60. 4x + 7 = 2(x – 3). Solve for x. A: 6.5 B: 6 C: -6.5 D: -5.5