Mathematics

1 Mathematics



Chapter outline


Basic Addition and Subtraction

Basic Multiplication (Whole Numbers)

Basic Division (Whole Numbers)

Decimals

Fractions

Multiplication of Fractions

Division of Fractions

Changing Fractions to Decimals

Changing Decimals to Fractions

Ratios and Proportions

Percentages

Regular Time versus Military Time

Algebra

Helpful Information to Memorize

Answers to Sample Problems


Key terms


Common Denominator


Constant


Denominator


Digit


Dividend


Divisor


Exponent


Expression


Factor


Fraction Bar


Improper Fraction


Least Common Denominator


Numerator


Percent


Place Value


Product


Proper Fraction


Proportion


Quotient


Ratio


Reciprocals


Remainder


Terminating Decimal


Variable


Members of the health professions use math every day to calculate medication dosages, radiation limits, nutritional needs, mental status, intravenous drip rates, intake and output, and a host of other requirements related to their clients. Safe and effective care is the goal of all who work in the health professions. Therefore it is essential that students entering the health professions be able to understand and make calculations using whole numbers, fractions, decimals, and percentages.


The purpose of this chapter is to review the addition, subtraction, multiplication, and division of whole numbers, fractions, decimals, and percentages. Basic algebra skills will also be reviewed: evaluating expressions, and solving for a specific variable. Mastery of these basic mathematic functions is an integral step toward a career in the health professions.



Basic Addition and Subtraction



Vocabulary


Digit: Any number 1 through 9 and 0 (e.g., the number 7 is a digit).


Place Value: The value of the position of a digit in a number (e.g., in the number 321, the number 2 is in the “tens” position).


image

(From Ogden SJ, Fluharty LK: Calculation of drug dosages: A work text, ed 9, St. Louis, 2012, Elsevier/Mosby.)



HESI Hint


1 ten = 10 ones


1 hundred = 100 ones


1 thousand = 1000 ones



Basic Addition








Example 1













462 + 133 462
  + 133
  595


Steps



1. Line up the digits according to place value.

2. Add the digits starting from right to left:
• Ones: 2 + 3 = 5

• Tens: 6 + 3 = 9

• Hundreds: 4 + 1 = 5


Addition with Regrouping




HESI Hint


To solve an addition problem, it may be necessary to regroup by moving, or carrying over, an extra digit from one place value column to the next.







Example 2
















835 + 559 1
  835
  + 559
  1,394


Steps



1. Line up the digits according to place value.

2. Add:
• Ones: 5 + 9 = 14
• Carry the 1 to the tens place, which is one place to the left.

• Tens: 1 + 3 + 5 = 9

• Hundreds: 8 + 5 = 13


Basic Subtraction


Subtraction provides the difference between two numbers.



HESI Hint


It may be easier to solve a subtraction problem by first rewriting it vertically.







Example 1













5,234 − 4,112 5,234
  − 4,112
  1,122


Steps



1. Line up the digits according to place value.

2. Subtract:
• Ones: 4 − 2 = 2

• Tens: 3 − 1 = 2

• Hundreds: 2 − 1 = 1

• Thousands: 5 − 4 = 1


Subtraction with Regrouping




HESI Hint


Remember, if the number to subtract is not a positive number, you must borrow, or regroup, from one place value to a lower place value.







Example 2













457 – 29 image
  –29
  428


Steps



1. Align the digits according to place value.

2. Subtract:
• Ones: 7 – 9
• Must borrow 1 from the 5 in the tens place: 17 – 9 = 8

• Tens: 4 – 2 = 2

• Hundreds: 4 – 0 = 4



Sample Problems


Add or subtract each of the following problems as indicated.


1. 1,803 + 156 =

2. 835 + 145 =

3. 1,372 + 139 =

4. 123 + 54 + 23 =

5. 673 – 241 =

6. 547 – 88 =

7. 222 – 114 =

8. 12,478 – 467 =

9. Jeff walks 5 miles west then turns north and walks 8 miles. How far has Jeff walked?

10. Julie picks 26 tomatoes from the tomato plants in her garden. She gives seven tomatoes to her next-door neighbor. How many tomatoes does Julie have now?


Basic Multiplication (Whole Numbers)


The process of multiplication is essentially repeated addition.



Vocabulary


Product: The answer to a multiplication problem.



HESI Hint


Remember, the placeholders help keep the problem aligned. If you do not skip a space, the answer will be incorrect. Below is an example of a well-aligned problem.



image








Example 1



















23 × 5  
  1
  23
  × 5
  115


Steps



1. Multiply one digit at a time.

2. Multiply 5 × 23.
• Ones: 5 × 3 = 15
Carry the 1 to the tens place, and write the 5 in the ones place.

• Tens: 5 × 2 = 10 + 1 = 11






Example 2







623 × 45 image


Steps



1. Multiply 623 × 5.
• 5 × 3 = 15

• 5 × 2 = 10 + 1 (carried over) = 11

• 5 × 6 = 30 + 1 (carried over) = 31 (does not need to be carried)

2. Multiply 623 × 4 (remember to line up the ones digit with the 4 by using zero as a placeholder).
• 4 × 3 = 12

• 4 × 2 = 8 + 1 = 9

• 4 × 6 = 24

3. Add the two products together.
• 3,115 + 24,920 = 28,035 (the final product)






Example 3






















301 × 451 301
  × 451
  301
  15,050
  + 120,400
  135,751


Steps



1. Multiply 301 × 1.
• 1 × 1 = 1

• 1 × 0 = 0

• 1 × 3 = 3

2. Multiply 301 × 5.
• 5 × 1 = 5 (remember to use a zero for a placeholder)

• 5 × 0 = 0

• 5 × 3 = 15

3. Multiply 301 × 4.
• 4 × 1 = 4

• 4 × 0 = 0

• 4 × 3 = 12

4. Add the three products together.
• 301 + 15,050 + 120,400 = 135,751 (the final product)



Sample Problems


Multiply each of the following problems as indicated.


1. 846 × 7 =

2. 325 × 6 =

3. 653 × 12 =

4. 806 × 55 =

5. 795 × 14 =

6. 999 × 22 =

7. 582 × 325 =

8. 9438 × 165 =

9. Jan is preparing an examination for 29 students. Each student will have 30 questions, with no student having duplicate questions. How many questions will Jan need to prepare?

10. John is ordering lunch for the volunteers at the hospital. There are 12 units in the hospital, with 15 volunteers in each unit. How many lunches will John need to order?


Basic Division (Whole Numbers)



Vocabulary


Dividend: The number being divided.


Divisor: The number by which the dividend is divided.


Quotient: The answer to a division problem.


Remainder: The portion of the dividend that is not evenly divisible by the divisor.



HESI Hint


image


The 45 represents the dividend (the number being divided), the 5 represents the divisor (the number by which the dividend is divided), and the 9 represents the quotient (the answer to the division problem). It is best not to leave a division problem with a remainder, but to end it as a fraction or decimal instead. To make the problem into a decimal, add a decimal point and zeros at the end of the dividend and continue. If a remainder continues to occur, round to the hundredths place.






Example:

233.547 → 233.55 (the 7 rounds the 4 to a 5)







Example 1







40 ÷ 8 image


Steps



1. Set up the problem.

2. Use a series of multiplication and subtraction problems to solve a division problem.

3. 8 × ? = 40
• Multiply: 8 × 5 = 40

• Subtract: 40 − 40 = 0

• The quotient (or answer) is 5.






Example 2







672 ÷ 6 image


Steps



1. Set up the problem.

2. Begin with the hundreds place.
• 6 × ? = 6. We know 6 × 1 = 6; therefore place the 1 (quotient) above the 6 in the hundreds place (dividend). Place the other 6 under the hundreds place and subtract: 6 − 6 = 0.

• Bring down the next number, which is 7; 6 × ? = 7. There is no number that can be multiplied by 6 that will equal 7 exactly, so try to get as close as possible without going over 7. Use 6 × 1 = 6 and set it up just like the last subtraction problem: 7 − 6 = 1.

• Bring down the 2 from the dividend, which results in the number 12 (the 1 came from the remainder of 7 − 6 = 1).

• 6 × ? = 12; ? = 2. The two becomes the next number in the quotient. 12 − 12 = 0. There is not a remainder.

• The quotient (or answer) is 112.






Example 3







174 ÷ 5 image


Steps



1. Set up the problem.

2. 5 does not divide into 1 but does divide into 17.

3. 5 × 3 = 15. Write the 3 in the quotient. (It is written above the 7 in 17 because that is the last digit in the number.)
• 5 × 3 = 15

• 17 − 15 = 2

4. Bring the 4 down. Combine the 2 (remainder from 17 − 15) and 4 to create 24.

5. Five does not divide evenly into 24; therefore try to get close without going over.
• 5 × 4 = 20

• 24 − 20 = 4

6. There is a remainder of 4, but there is not a number left in the dividend. Add a decimal point and zeros and continue to divide.

7. The quotient (or answer) is 34.8 (thirty-four and eight tenths).



Sample Problems


Divide in each of the following problems as indicated.


1. 132 ÷ 11 =

2. 9,618 ÷ 3 =

3. 2,466 ÷ 2 =

4. 325 ÷ 13 =

5. 5,024 ÷ 8 =

6. 3,705 ÷ 5 =

7. 859 ÷ 4 =

8. 6,987 ÷ 7 =

9. There are 225 pieces of candy in a large jar. Ben wants to give the 25 campers in his group an even amount of candy. How many pieces of candy will each camper receive?

10. Edie has 132 tulip bulbs. She wants to plant all of the tulip bulbs in 12 rows. How many tulip bulbs will Edie plant in each row?


Decimals


A decimal pertains to tenths or to the number 10.



Vocabulary


Place value: Regarding decimals, numbers to the right of the decimal point have different terms from the whole numbers to the left of the decimal point. Each digit in a number occupies a position called a place value.



HESI Hint


Remember, whole numbers are written to the left of the decimal point and place values are written to the right of the decimal point. Line the numbers up vertically before solving the problem.



Addition and Subtraction of Decimals



image

(From Ogden SJ, Fluharty LK: Calculation of drug dosages: A work text, ed 9, St. Louis, 2012, Elsevier/Mosby.)



HESI Hint


The word “and” stands for the decimal when writing a number in words.


Example: 5.7 (five and seven tenths)







Example 1













2.6 + 3.1 2.6
  + 3.1
  5.7


Steps



1. Align the decimal points.

2. Add the tenths together: 6 + 1 = 7

3. Add the ones together: 3 + 2 = 5

4. Final answer: 5.7 (five and seven tenths).






Example 2













5 + 12.34 12.34
  + 5.00
  17.34


Steps



1. Align the decimal points.
• It might be difficult to align the 5 because it does not have a decimal point. Remember that after the ones place, there is a decimal point. To help with organization, add zeros (placeholders). Example: 5 = 5.00

2. Add the hundredths: 4 + 0 = 4

3. Add the tenths: 3 + 0 = 3

4. Add the ones: 2 + 5 = 7

5. Add the tens: 1 + 0 = 1

6. Final answer: 17.34 (seventeen and thirty-four hundredths).






Example 3













7.21 − 4.01 7.21
  − 4.01
  3.20


Steps



1. Align the decimal points.

2. Subtract the hundredths: 1 − 1 = 0

3. Subtract the tenths: 2 − 0 = 2

4. Subtract the ones: 7 − 4 = 3

5. Final answer: 3.20 (three and twenty hundredths).






Example 4







12 − 8.99 image


Steps



1. Align the decimal points.

2. Because 12 is a whole number, add a decimal point and zeros.

3. 0.00 − 0.99 cannot be subtracted; therefore 1 must be borrowed from the 12 and regrouped.

4. The ones become 1, the tenths become 9, and the hundredths become 10.

5. Subtract the hundredths: 10 − 9 = 1

6. Subtract the tenths: 9 − 9 = 0

7. Subtract the ones: 11 − 8 = 3
• 1 was borrowed from the tens in order to subtract the 8.

8. Final answer: 3.01 (three and one hundredth).



Sample Problems


Solve each of the following decimal problems as indicated.


1. 9.2 + 7.55 =

2. 2.258 + 64.58 =

3. 892.2 + 56 =

4. 22 + 3.26 =

5. 8.5 + 7.55 + 14 =

6. 18 – 7.55 =

7. 31.84 – 2.430 =

8. 21.36 – 8.79 =

9. Bill has 2.5 vacation days left for the rest of the year and 1.25 sick days left. If Bill uses all of his sick days and his vacation days, how many days will he have off work?

10. Erin has 6.25 peach pies. She gives Rose 3.75 of the peach pies. How many pies does Erin have left?


Multiplication of Decimals



image

(From Ogden SJ, Fluharty LK: Calculation of drug dosages: A work text, ed 9, St. Louis, 2012, Elsevier/Mosby.)







Example 1



image


Steps



1. Multiply 757 × 21 (do not worry about the decimal point until the final product has been calculated).

2. Starting from the right, count the decimal places in both numbers and add together (two decimal places).

3. Move to the left two places, and then place the decimal point.






Example 2



image


Steps



1. Multiply 2 × 34.

2. Starting from the right, count the decimal places in both numbers and add together (four decimal places).

3. Move to the left four places, and then place the decimal.






Example 3



image


Steps



1. Multiply 341 × 7.

2. Starting from the right, count the decimal places in both numbers and add together (two decimal places).

3. Move to the left two places, and then place the decimal point.



Sample Problems


Multiply the decimals in the following problems as indicated.


1. 0.003 × 4.23 =

2. 98.26 × 8 =

3. 8.03 × 2.1 =

4. 250.1 × 25 =

5. 0.1364 × 2.11 =

6. 8.23 × 4 =

7. 0.058 × 64.2 =

8. 794.23 × .001 =

9. Jenny lost 3.2 lb each month for 6 months. How much weight has Jenny lost?

10. Richard wants to make 2.5 batches of sugar cookies. One batch calls for 1.75 cups of sugar. How many cups of sugar will Richard need for 2.5 batches of cookies?


Division of Decimals



image

(From Ogden SJ, Fluharty LK: Calculation of drug dosages: A work text, ed 9, St. Louis, 2012, Elsevier/Mosby.)



HESI Hint


The number 25 is a whole number. Though this number could be written 25.0, decimals are usually not displayed after a whole number.







Example 1









34 ÷ 2.5 image
 


Steps



1. Set up the division problem.

2. Move the decimal point in 2.5 one place to the right, making it a whole number.

3. What is done to one side must be done to the other side. Move the decimal point one place to the right in 34, making it 340, and then bring the decimal point up into the quotient.

4. Divide normally.
• 25 × 1 = 25

• Subtract: 34 − 25 = 9

• Bring down the zero to make 90.

• 25 × 3 = 75. This is as close to 90 as possible without going over.

• Subtract: 90 − 75 = 15

• Add a zero to the dividend and bring it down to the 15, making it 150.

• 25 × 6 = 150

• 150 − 150 = 0

5. The quotient is 13.6.






Example 2







2.468 ÷ 0.2 image


Steps



1. Set up the division problem.

2. Move the decimal point in 0.2 over one place to the right, making it a whole number. 0.2 is now 2.

3. Move the same number of spaces in the dividend. 2.468 is now 24.68.

4. Bring the decimal point up to the quotient in the new position.

5. Divide normally.






Example 3











0.894 ÷ 0.05 image
 
 


Steps



1. Set up the division problem.

2. Move the decimal point in the divisor until it is a whole number. 0.05 is now 5.

3. Move the decimal point in the dividend the same number of spaces as was moved in the divisor. 0.894 is now 89.4.

4. Divide normally.



Sample Problems


Divide the decimals in the following problems as indicated.


1. 48 ÷ 0.4 =

2. 144 ÷ 0.6 =

3. 3.75 ÷ 0.4 =

4. 56.2 ÷ 0.2 =

5. 2.6336 ÷ 0.32 =

6. 591 ÷ 0.3 =

7. 0.72 ÷ 0.8 =

8. 0.132 ÷ 0.11 =

9. Stewart has 56 acres of land. He wants to divide the land into plots of 0.25 acres. How many plots of land will Stewart have after he divides the 56 acres?

10. Donna has 4.2 liters of fertilizer. If each pecan tree needs 0.7 liters of fertilizer and Donna uses all of the fertilizer, how many pecan trees does Donna have?


Fractions


In mathematics, a fraction is a way to express a part in relation to the total.



Vocabulary


Numerator: The top number in a fraction.


Denominator: The bottom number in a fraction.


Fraction Bar: The line between the numerator and denominator. The bar is another symbol for division.


Factor: A number that divides evenly into another number.


Least Common Denominator: The smallest multiple that two numbers share.


Improper Fraction: A fraction where the numerator is larger than the denominator.


Proper Fraction: A fraction where the denominator is larger than the numerator.


Common Denominator: Two or more fractions having the same denominator.


Reciprocals: Pairs of numbers that equal 1 when multiplied together.


Terminating Decimal: A decimal that is not continuous.



image




HESI Hint



• The numerator is the top number of the fraction. It represents the part or pieces.

• The denominator is the bottom number of the fraction. It represents the total or whole amount.

• The fraction bar is the line that separates the numerator and the denominator.


Reducing Fractions Using the Greatest Common Factor


A factor is a number that divides evenly into another number.


Factors of 12:


• 1 × 12 = 12

• 2 × 6 = 12

• 3 × 4 = 12

12 {1, 2, 3, 4, 6, 12}: Listing the factors helps determine the greatest common factor between two or more numbers.



image



All represent one-half.


Reducing fractions can also be called reducing a fraction to its lowest terms or simplest form.



image








Example 1

Reduce image


Factors of 7 and 21:


7 {1, 7}


21 {1, 3, 7, 21}


The greatest common factor is 7; therefore divide the numerator and denominator by 7.



image


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Apr 10, 2017 | Posted by in NURSING | Comments Off on Mathematics

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