There are some mathematical concepts that we will use so often in this class that it is important that we take the time to review them now, before we begin any new material. Each of the mathematical ideas explained below should be something that you've done before in a class that is a prerequisite to this one; however, we are covering them briefly here because you will be unable to do any of the problems in this class if you do not remember these important mathematical techniques from previous classes.

So, for example, the number 504.205 has 5 hundreds, 4 ones, 2 tenths, and 5 thousandths. To compare the relative sizes of two decimals, we will need be able to identify each place value in the decimal. To see which decimal is bigger, we go from **left to right**; we begin by looking at the place value **farthest to the left** and comparing its digit in each number. If the place value **farthest to the left** is the **same** in each number, then we look at the **next place value to the right**, and if that digit is the same in each number, we keep going to the next place value to the right until we find a digit that is different in each number. (If all the digits are the same in every place value, then the two numbers we are comparing must be exactly the same!)

1 |

100 |

1 |

10 |

1 |

100 |

1 |

10 |

Now it's time for you to practice some problems of your own.

- If the number to the
**right**of the place where we want to round is**less than 5**, then we**round down**. This means that we leave the number in the place where we want to round the same; then we leave off any numbers in decimal places to the right of this spot. - If the number to the
**right**of the place where we want to round is**greater than or equal to 5**, then we**round up**. This means that we take the number in the place where we want to round and increase it by one; then we leave off any numbers in decimal places to the right of this spot.

Sometimes when the number we are trying to **round up** is a **9**, we actually have to round the **9** up to a **0**, and then **increase the number directly to the left of it by one**; this is like rounding the 9 up to a 10 - we just can't put the number 10 in a single place - we can only write it by going up by one, one place value to the left.

Round 3.45 to the nearest tenth.

We want this number to stop at the **tenths** place, so we look at the number just to the right of the tenths place, the number in the **hundredths** place:

Because the number in the **hundredths** place is 5, we **round up**, so we change the 4 in the **tenths** place to a 5 and leave all numbers that come after the 4 off when we right the rounded number:

3.5

Round 3.452 to the nearest hundredth.

We want this number to stop at the **hundredths** place, so we look at the number just to the right of the tenths place, the number in the **thousandths** place:

Because the number in the **thousandths** place is 2, we **round down **, so we leave the 5 in the **hundredths** as a 5 and leave all numbers that come after the 5 off when we write the rounded number:

3.45

Round 1.99 to the nearest tenth.

We want this number to stop at the **tenths** place, so we look at the number just to the right of the tenths place, the number in the **hundredths** place:

Because the number in the **hundredths** place is 9, we **round up**, so we change the 9 in the **tenths** place to a 0, and change the number 1, which is one place to the **left** of the tenths place, to a 2; then we leave all numbers that come after the tenths place off when we write the rounded number:

2.0=2

In this course we will always want to **eliminate extra zeros after the decimal point** because when working with drug dosages, misreading the number or the units of the dosage is one of the most common reasons for patients being dosed incorrectly. Because 2.0 may sometimes be misread as 20, even though it is the more "mathematically correct" way of writing the answer, we will always write 2.0 as 2 instead, because we are more interested in making sure that the dosage will not be misread.

**Line up the decimal points**.- Add as usual,
**keeping the decimal point of the answer lined up with the decimal points of the numbers we are adding.**

34.5+7

First we line up the decimal points:

34.5Then we add and carry as needed:

134.5

41.5

2.96+19.3

First we line up the decimal points:

2.96Then we add and carry as needed:

1 12.96

22.26

**Line up the decimal points**.**Add zeros after the decimal point**to the**first**number in the subtraction problem; you will need the first number to have at least the same number of digits after the decimal point as the second number so that you can subtract!- Subtract as usual,
**keeping the decimal point of the answer lined up with the decimal points of the numbers we are adding.**

Also,

34.5−7

First we line up the decimal points:

34.5Then we subtract and borrow as needed:

227.5

20−9.99

First we line up the decimal points:

20Then we put the decimal point and 2 zeros after 20 because we recall that 20=20.00:

20.00And finally we subtract and borrow as needed:

1_{10}

~~2~~0.00 We need to borrow to subtract 9 from 0, so we keep going to the left until we can borrow -

__− 9.99 __ we borrow from the 2.

_{9}

1_{10}_{10}

~~2~~0.00 Now we keep borrowing until we can subtract the 9 from the 0 on the far right-

__− 9.99 __ we borrow from the 10.

_{9 9}

1_{10}_{1010}

~~2~~0.00 Now we keep borrowing until we can subtract the 9 from the 0 on the far right-

__− 9.99 __ we need to borrow from the second 10, and then we can subtract.

10.01

- Multiply as usual, ignoring the decimal points for the moment. Be sure to
**leave any zeros at the end**! **Count the number of digits after the decimal point**in the two numbers you are multiplying.- Take the number you got in step 2 and move the decimal place
**that many places from the right**in your answer.**Zeros count as a place**when you are moving the decimal in this step!

34.5×7

First we multiply, ignoring the decimal points:

3334.5

2415

Then we count the number of digits after the decimal place in the numbers we are multiplying - in this case there is only 1 digit total (one in 34.5 and none in 7):

3334.5

2415

So now we count 1 digit from the right in our answer, and we put the decimal place there:

241.6.25×32.4

First we multiply, ignoring the decimal points:

11

1 2

6.25

2500

1250

202500

Then we count the number of digits after the decimal place in the numbers we are multiplying - in this case there are 3 digits total (two in 6.25 and one in 32.4):

6.25202500

So now we count 3 digits from the right in our answer, and we put the decimal place there:

202.__5____0____0__

Notice that we can eliminate any unnecessary zeros **after the decimal point at the end**, so this becomes:

- Turn both decimals into whole numbers by multiplying each one by a power of 10. As long as you multiply
**both**numbers by the**same**power of ten, you will not change the answer to the division problem. - Make sure to put the correct number under the long division sign. The
**first**number is the number that is**getting divided up**and the**second**number is the number that is**doing the dividing**. - Put a decimal point above the decimal point in the number below the division bar.
- Divide as usual - if you need to, you can add as many zeros as you like
**after**the decimal point to the number below the fraction bar. - Keep dividing until you get a zero remainder, or until you have
**one digit beyond**the place you need to round to. (For example, if you are rounding to the tenths place, you cannot stop dividing after you have the tenths place because you will not know whether you should round up or down - to round to the**tenths**place you must keep dividing until you have the**hundredths**place. )

34.5 |

7 |

First we multiply by a power of 10 to turn both numbers into whole numbers. Because 34.5 is the only decimal here and we only need to move the decimal place over **one spot to make it a whole number, we multiply both numbers by 10**:

34.5×10 |

7×10 |

345 |

70 |

Now we set up our long division. Because 345 is on the **top**, it is what is being divided up, so it goes **under** the division sign, and because 70 is on the **bottom**, it is what is doing the dividing, so it goes to the **left** of the division sign:

70|345

So now we add zeros after the decimal point in 345; since 345 is a whole number, we have to write in the decimal point after the 5, and then we can add as many zeros at the end as we like. We also write in a decimal point above the decimal point in 345:

.

70|345.000

So now we divide as usual. We must keep going until we get to the **thousandths** place because we need to round to the **hundredths** place :

4.928

70|345.000

__−280__ ↓ ↓ ↓

650 ↓ ↓

__−630__ ↓ ↓

200 ↓

__−140__ ↓

600

So now we round 4.928 to the **hundredths** place; since 8 is the number in the **thousandths** place and it is greater than or equal to 5, we can then round it to:

4.93

6.25 |

32.4 |

First we multiply by a power of 10 to turn both numbers into whole numbers. Because 6.25 has **two** numbers after the decimal place (and 32.4 has only one number after the decimal place, we multiply **both** numbers by **100**:

6.25×100 |

32.4×100 |

625 |

3240 |

Now we set up our long division. Because 625 is on the **top**, it is what is being divided up, so it goes **under** the division sign, and because 3240 is on the **bottom**, it is what is doing the dividing, so it goes to the **left** of the division sign:

So now we add zeros after the decimal point in 625; since 625 is a whole number, we have to write in the decimal point after the 5, and then we can add as many zeros at the end as we like. We also write in a decimal point above the decimal point in 625:

.

3240|625.000

So now we divide as usual. We must keep going until we get to the **thousandths** place because we need to round to the **hundredths** place :

0.192

3240|625.000

__−3240__ ↓ ↓

30100 ↓

__−29160__ ↓

9400

So now we round 0.192 to the **hundredths** place; since 2 is the number in the **thousandths** place and it is less than 5, we can then round it to:

0.19

Because this class is specifically concerned with calculating dosages, we will always be doing word problems. So in order to sucessful in setting up our word problems, we need to practice.

The math operations we use in this class are very simple: addition, subtraction, multiplication, or division. So all we need to do is to try to determine when to do each of these operations. Let's remind ourselves what each of these operations are for:

- A problem that talks about finding the
**total**of something, or describes**combining**or**putting things together**is telling us to add, because addition is the process of putting things together. - A problem that talks about finding the
**difference**between two amounts, or finding out**what is left**after something is taken away is telling us to subtract, because subtraction is the process of taking one thing from another, or finding the difference between two things. - A problem that talks about doing something multiple
**times**, especially when putting things together to find a total is telling us to multiply, because multiplication is the process of adding or combining things multiple times. - A problem that talks about
**breaking a total or a whole down into smaller pieces**is asking us to divide, because division is the process of taking a whole and breaking it into several equal pieces.

This morning you gave a patient 3.5 mg of a drug, and then this afternoon you gave them 4 mg; how much of the drug has the patient taken today?

Notice that this question is asking for the **total** amount of the drug the patient has taken after a few dosings - this clearly points to **addition**, so the problem is written:

3.5 mg +4 mg

This equals:

7.5 mg

You need to give a patient a dosage of 2.8 mg, but you only have one tablet labeled 0.7 mg. How many more milligrams do you need to give the patient this dosage?

Notice that in this question, they are looking for how many milligrams we have **left** to find after we take into account the part of the total dosage that can be made up by the 0.7 mg tablet. So we are taking 0.7 mg away from the total 2.8 mg to see what is **left**. This process indicates **subtraction**, so the problem is written:

2.8 mg − 0.7 mg

This equals:

2.1 mg

Prepare a dosage of 3.2 mg using tablets with a strength of 1.6 mg. How many tablets do you need to give this dosage?

In this problem, you are asked to take a **total** amount, 3.2 mg, and **break it down into equal pieces** - separate tablets, each with a strength of 1.6 mg. This process of taking a whole or a total and finding the number of pieces it will break down into is **division**, so the problem should read:

3.2 mg |

1.6 mg |

Which equals:

2 tablets

The tablets available are labeled 10.5 mg and you are to give 3

1 |

2 |

Here you are finding a **total** dosage, so your first instinct might be to add, which you could do if you added each tablet separately: 10.5 mg + 10.5 mg + 10.5 mg + 5.25 mg

However, since we are giving the same amount (the 10.5 mg that is in each tablet) **multiple** times, it makes sense to **multiply**, so that the problem looks like this:

10.5 mg × 3.5 mg

This equals:

36.75 mg

- To multiply fractions,
**multiply straight across the top and straight across the bottom.** - If you have any decimals in your fraction, you can get rid of them by multiplying
**both the top and the bottom**of the fraction by a power of 10. - We can only cancel, or divide by a number, if we cancel it out
**once on the top**and**once on the bottom**. - Keep cancelling until you cannot cancel anything else. You will often have to cancel more than once!

1.3 |

0.75 |

0.9 |

1 |

First we get rid of the decimals. In this case, we want to turn 0.75 in the bottom into 75. We can do this by multiplying by 100, but if we multiply the bottom by 100, we must also multiply the top by 100:

1.3×100 |

0.75×100 |

0.9 |

1 |

130 |

75 |

0.9 |

1 |

Now we need to get rid of the decimal 0.9, so we want to turn 0.9 in the top into 9. We can do this by multiplying by 10, but if we multiply the top by 10, we must also multiply the bottom by 10:

130 |

75 |

0.9×10 |

1×10 |

130 |

75 |

9 |

10 |

Now we can work on simplifying the numbers 130, 75, 9 and 10 in the fractions by multiplying straight across to get one big fraction:

130×9 |

75×10 |

Now we can "cancel" things out in the top and the bottom by dividing numbers in the numerator and the denominator of the fraction by the **same number**. In this problem, we can divide both 130 and 10 by 10, and we can divide both 9 and 75 by 3:

13^{3} |

_{25} |

13×3 |

25×1 |

13×3 |

25 |

Now we should continue to cancel until we cannot cancel anymore, but we notice that we cannot cancel anymore because the only number that goes into 25 is 5, but 5 does not go into 13 or 3.

So, since we cannot cancel anymore, we will multiply across the top and the bottom:

13×3 |

25 |

39 |

25 |

Now all that is left for us to do is to divide 39 by 25, and then round off our answer. Here we will round to the nearest tenth. (In any assignments you receive in this class, the instructions will always tell you what place to round to, if you do not get an exact answer.):

39÷25=1.56

So now we round 1.56 to the nearest tenth to get:

1.6

44 |

1 |

1 |

2.2 |

2.1 |

1 |

First we get rid of the decimals. In this case, we want to turn 2.1 in the top into 21. We can do this by multiplying by 10, but if we multiply the top by 10, we must also multiply the bottom by 10:

44 |

1 |

1 |

2.2 |

2.1×10 |

1×10 |

44 |

1 |

1 |

2.2 |

21 |

10 |

Now we want to turn 2.2 in the bottom into 22. We can do this by multiplying by 10, but if we multiply the bottom by 10, we must also multiply the top by 10:

44 |

1 |

1×10 |

2.2×10 |

21 |

10 |

44 |

1 |

10 |

22 |

21 |

10 |

Now we can work on simplifying the numbers in the fractions by multiplying straight across to get one big fraction:

44×10×21 |

1×22×10 |

Now we can "cancel" things out in the top and the bottom by dividing numbers in the numerator and the denominator of the fraction by the **same number**. In this problem, we can divide both 10 and 10 by 10, and we can divide both 44 and 22 by 22:

^{2}×1 |

1×_{1} |

2×1×21 |

1×1×1 |

2×21 |

1 |

Now we should continue to cancel until we cannot cancel anymore, but we notice that we cannot cancel anymore because there is nothing left in the bottom except 1.

So, since we cannot cancel anymore, we will multiply across the top and the bottom:

2×21=42

7.5 |

12.3 |

55 |

5 |

23.2 |

1.2 |

First we get rid of the decimals. In this case, we want to turn 7.5, 12.3, 23.2, and 1.2 into whole numbers; because each of this will be a whole number if we multiply them by 10, we take both the first and the last fraction above and multiply **both the top and the bottom** of each fraction by 10:

7.5×10 |

12.3×10 |

55 |

5 |

23.2×10 |

1.2×10 |

75 |

123 |

55 |

5 |

232 |

12 |

Now we can work on simplifying the numbers in the fractions by multiplying straight across to get one big fraction:

74×55×232 |

123×5×12 |

Now we can "cancel" things out in the top and the bottom by dividing numbers in the numerator and the denominator of the fraction by the **same number**. In this problem, we can divide both 75 and 3 by 3, and we can divide both 232 and 12 by 34:

75×^{11}×^{58} |

123×_{1}_{3} |

75×11×58 |

123×1×3 |

Now we can reduce this fraction even further, by dividing 75 and 3 by 3:

^{25}×11×58 |

123×1×_{1} |

25×11×58 |

123×1×1 |

25×11×58 |

123 |

Now we should continue to cancel until we cannot cancel anymore, but we notice that we cannot cancel anymore because the only number that goes into 25 is 5, but 5 does not go into 123; likewise, 11 does not go into 123, and 2 and 29 are the only two numbers which go into 58, but neither of these two numbers goes into 58.

So, since we cannot cancel anymore, we will multiply across the top and the bottom:

25×11×58 |

123 |

15950 |

123 |

Now all that is left for us to do is to divide 15,950 by 123, and then round off our answer. Here we will round to the nearest whole number. (In any assignments you receive in this class, the instructions will always tell you what place to round to, if you do not get an exact answer.):

15,950÷123=129.6... (this decimal keeps going on forever, but we can stop dividing once we know what number is in the tenths place, because we will round to the ones place)

So now we round 129.6 to the nearest whole number to get:

130