BMS 1110 · Common Core (Tronc commun)Section 1.4 · Number foundations: algebra and fractions

Section 4 of 30

Decimals and percents

Read, compare, add, subtract, multiply and divide decimal fractions; move the point when multiplying or dividing by 10, 100, 1000 or by 0.1 and its multiples; round to a named place value; convert between decimals, fractions and percents; and answer the three percent questions.

New to this course? How each section works

Learn gives you one short idea at a time. Each idea starts with a single sentence, then explains it, gives an example, and asks you a small question you can answer straight away.
Explore shows a diagram of the same ideas. Click the numbered parts to read what each one is.
Check asks a few questions. Answer them all correctly to finish the section and unlock the next one.
“Go deeper” is optional. It adds mechanism and extra detail for when you are ready — nothing you need for the checks is hidden there.

Why this matters

Almost every number a nurse handles is a decimal or a percent: 0.5 mg of a drug, 1.25 g of a cream, 0.9% sodium chloride, a 2% solution. The decimal point is what carries the value, so a point written in the wrong place — or a leading zero left off — changes a dose tenfold while the digits stay exactly the same. Lining the points up, and moving them deliberately, is the single habit that keeps decimal arithmetic safe at the bedside.

What you need to know first

This section assumes only that you can add, subtract, multiply and divide whole numbers, and that you have already met a fraction as one number written over another. Everything else — place value, the moving point, rounding and percentages — is rebuilt here from the beginning, so no previous lesson is required.

Words to know (5)

Decimal fraction: A number written with a decimal point, in which the digits to the right of the point form a numerator over a denominator of 10, 100, 1000 and so on. 0.86 is a decimal fraction: the 86 sits over 100, so 0.86 = 86/100.
Place value: The value a digit has because of where it sits: tenths in the first place after the point, hundredths in the second, thousandths in the third. In 0.623 the 6 means six tenths, because it is the first digit after the point; the 2 means two hundredths.
Leading zero: The zero written to the left of the decimal point when a number is smaller than one, as in 0.5. Leaving it out is a recognised medication-safety hazard. 0.5 mg is read safely as half a milligram; .5 mg can be misread as 5 mg, ten times the dose.
Percent: A value equal to a number of hundredths. The word means “per hundred”, so a percent written as a fraction always carries 100 as its denominator. 45% = 45/100 = 0.45, and a percent containing a fraction, 12½%, is 0.125.
Dividend and divisor: In a division, the dividend is the number being divided and the divisor is the number you divide by. The quotient is the answer the division produces. In 0.25 ÷ 0.01 = 25, the dividend is 0.25, the divisor is 0.01 and the quotient is 25.

One idea at a timeIdea 1 / 7

In one sentence A decimal fraction names its denominator by position: the first place after the point is tenths, then hundredths, then thousandths.

The decimal point carries the value of every digit

The decimal point is the only mark that tells you what each digit is worth. Every decimal fraction is a numerator written in numerals, with a point marking a denominator of 10 or a power of 10. In 0.623 the point announces that the digits to its right are tenths, hundredths and thousandths: 6 tenths, 2 hundredths and 3 thousandths, so 0.623 = 6/10 + 2/100 + 3/1000. Whole numbers stand to the left of the point and decimal parts to the right. Move the point and you have changed the quantity, even though the digits are untouched. That is why a zero must always be written to the left of the point when the number is smaller than one: .5 and 5 look alike on a hurried chart or a smudged order, and the second is ten times the first. Write 0.5, never .5.

For example

A stock label reads 0.0096 g. Reading from the point, the 9 is thousandths and the 6 is ten-thousandths, so the amount is nine thousand six hundred millionths of a gram — a quantity only a careful place-value reading gets right.

Pause and check yourself

What does the digit 2 mean in 0.623, and what value would the digits have if the point were moved one place to the right?

Show the answer

The 2 is in the hundredths place, so it means 2/100. Moving the point one place to the right turns 0.623 into 6.23, so every digit becomes ten times larger: the 2 now means 2/10, or two tenths.

Read detailed explanations

Leading and trailing zeros: a safety rule with a mechanism: Leading and trailing zeros are not handwriting preferences; they are dose-safety devices, and the mechanism behind them is worth understanding. A decimal point is a single mark, easily lost in handwriting, in a scanned prescription or in an interrupted keyboard entry. When the zero in front of the point is missing, the digits that remain read as a whole number: .5 mg becomes 5 mg, a tenfold increase, and .25 mg becomes 25 mg, a hundredfold increase. The hazard is doubled because the erroneous reading is a perfectly ordinary dose, so nothing about the number looks strange to a second checker. The reverse hazard sits after the point. A trailing zero — writing 5.0 mg when the measured amount is 5 mg, or 1.50 g instead of 1.5 g — suggests that the zero carries information it does not have, and it creates a landing place for a point that can be dropped or displaced. The internationally taught convention is therefore symmetrical: always a leading zero, never a trailing zero unless the zero is genuinely significant for the preparation or the measuring device. Reading a label applies the same discipline in reverse. Percentage strengths, microgram quantities in small print and ratio expressions such as 1:1000 all present the same ambiguity, and the verification habit is to read the quantity aloud with its unit, then confirm that the volume of the container and the size of the dose agree. A decimal point is the cheapest piece of safety equipment in the medication room, and it is worth the extra half-second to write the zero in front of it.
Base ten: why moving the point is the whole rule: Place value is a base-ten system, and that single fact explains why powers of ten move the point instead of changing the digits. Write 0.132 as 1 × 10⁻¹ + 3 × 10⁻² + 2 × 10⁻³. Multiplying by 10 adds one to every exponent: 1 × 10⁰ + 3 × 10⁻¹ + 2 × 10⁻², which is 1.32. Multiplying by 100 adds two to every exponent, and multiplying by 1000 adds three. Nothing about the digits changes; the power attached to each digit changes, and we see that as the point sliding right. Dividing by 10, 100 or 1000 subtracts from each exponent and slides the point left. This is the same architecture as the metric system used for every weight and volume: 1000 mg = 1 g means the gram sits three places to the left of the milligram, so converting between them is a point movement, not a separate formula to memorise. The reciprocal facts follow immediately. Because 0.1 = 10⁻¹ and its reciprocal is 10, dividing by 0.1 is multiplying by 10; dividing by 0.01 is multiplying by 100; dividing by 0.001 is multiplying by 1000. Learners who memorise “the point moves right” without the exponent picture eventually meet a case such as 0.729 × 0.1, where the direction seems to contradict the rule for multiplication; with the exponents, 0.1 means one place smaller, so 0.729 becomes 0.0729. The rule has one source, and that source is the base-ten exponent.
Percent, fraction, decimal and ratio: one quantity in four notations: Percent, fraction, decimal and ratio are four notations for one quantity, and the three percent questions in the syllabus are one relationship seen from three sides. Take the sentence “9 is 37.5% of 24”. The base, 24, is the quantity you are taking a part of; the part, 9, is what you have; and the rate, 37.5%, expresses that part as so many hundredths of the base. Written as an equation, part = rate × base, and every question is answered by rearranging that single equation. “What percent of 24 is 9?” asks for the rate: 9 ÷ 24 = 0.375 = 37.5%. “What is 37.5% of 24?” asks for the part: 0.375 × 24 = 9. “9 is 37.5% of what number?” asks for the base: 9 ÷ 0.375 = 24. Clinical work adds two vocabulary habits. First, a percentage strength is a concentration, not a bare proportion: 0.9% w/v means 0.9 g in 100 mL, and the denominator 100 carries the unit with it, so reading it as “0.9 grams per hundred millilitres” prevents the frequent mistake of treating 0.9% as 0.9 g per litre. Second, percentage points and percent change are different quantities and must not be substituted for one another: a reduction from 4% to 3% is a fall of one percentage point, which is at the same time a relative fall of 25%. Confusing the two misreports audits and incident data by a wide margin, and the error usually survives because both numbers are small and both sound plausible.

Explore the diagram

Look at the whole drawing first. Then select one numbered part and connect it to the idea you just read.

Decimals and percents: the point carries the value

Four teaching panels. The first shows the place-value columns of 0.623 with the fraction each digit names, and the mandatory leading zero. The second shows that multiplying by 100 and dividing by 0.01 both move the point two places right, while dividing by a power of ten moves it left. The third walks a 1% strength through a decimal to a 25 mL dose, naming the dividend and the divisor. The fourth collects rounding, the fraction-to-decimal division and the percent-to-decimal conversion.

New to diagrams? Read this first

Look at the whole drawing once without clicking anything. Ask yourself only: what kind of picture is this (a comparison, a route, a sequence, a table)?
Press Play, or use the arrows, to move through the numbered parts in order. The drawing stays still; only the highlight moves.
Click a number, or its name in the list below, to read what that part is and why it matters. Use “Hide labels” when you want to test your memory.

More controls

100%

01 / 4Place value fixes what each digit is worth

If the drawing is wider than this panel, scroll it sideways. Labels stay readable; the numbered explanations below show the same ideas.

Place value fixes what each digit is worth

The decimal point is the only mark that gives the digits their value. In 0.623 the 6 sits in the tenths column, the 2 in the hundredths and the 3 in the thousandths, so the number reads as 6/10 + 2/100 + 3/1000. Move the point one place and the same digits mean ten times more. The red point marks the separator; the zero to its left is not decoration, because .5 without it can be read as 5 and becomes a tenfold dose error.

Schematic teaching plate drawn to a regular layout; the arrows show the direction the decimal point travels, not a measured distance. The 1% w/v strength is the worked example and is not a dose recommendation.

Key points to remember

The decimal point carries the value: tenths, hundredths and thousandths are named by position, and a zero must always be written to its left as in 0.5.
To compare, add or subtract decimals, line the points up in a column and pad with zeros on the right until every number has the same number of places.
Multiply the digits, then restore as many decimal places as the two factors had together; to divide by a decimal, shift both points right until the divisor is whole.
×10, ×100 and ×1000 move the point one, two and three places right; ÷0.1, ÷0.01 and ÷0.001 do the same, because dividing by a hundredth is multiplying by a hundred.
Percent means hundredths, so percent-to-decimal divides by 100 and decimal-to-percent multiplies by 100; part = percent × whole solves all three percent questions.

Go deeper

Worked example

Finding 12½% of a dose without losing the point. A cream contains 12½% of an active drug, and the pharmacy needs to know how much drug is present in a 240 mg quantity of the cream. The percent contains a fraction, so the decimal must be built carefully before any multiplication happens.

Write the percent with its fractional part as a decimal: 12½% becomes 12.5%.
Remember that percent means hundredths, so place 12.5 over 100: 12.5/100.
Divide by 100, which moves the point two places left: 12.5 ÷ 100 = 0.125.
Multiply the decimal by the quantity: 0.125 × 240 mg.
Calculate and attach the unit: 0.125 × 240 = 30, so 12½% of 240 mg is 30 mg of active drug.

Every percent question follows these moves: name the percent, write it as hundredths, divide by 100 to get the decimal, then multiply by the whole. The unit of the whole becomes the unit of the answer, and the size of the answer should always be a sensible fraction of the whole.

Tutorial 1 — a percentage strength turned into a dose

A ward stocks a drug as a 1% w/v solution, which the label explains as 1 g of drug in every 100 mL. A patient is ordered 250 mg. Work out how many millilitres you must draw up, converting the percentage strength to a decimal on the way.

Rule used: A strength written as a percent is a number of grams per 100 mL, so concentration in g/mL = percent ÷ 100. Then volume to give in mL = dose required in g ÷ concentration in g/mL, and the dose required is always the dividend.

Write the strength as a fraction with 100 as its denominator, then as a decimal of grams per millilitre.
Reveal this step
1% = 1/100 = 0.01, so the solution contains 0.01 g of drug in every 1 mL. The point moved two places left, exactly as percent-to-decimal always does.
How many grams of drug are in 100 mL, and how does that become 0.01 g per mL?
Reveal the answer
The label says 1%, so there is 1 g of drug in 100 mL. Dividing 1 g by 100 mL gives 0.01 g in each 1 mL, which is the concentration we will divide by.
Convert the ordered dose from milligrams to grams so that both quantities are expressed in the same unit.
Reveal this step
250 mg = 0.25 g, because 1 g = 1000 mg and 250 ÷ 1000 = 0.25. The point moved three places left, from 250 to 0.25.
Which number is the dividend and which is the divisor when you calculate the volume, and what are their units?
Reveal the answer
The dose required, 0.25 g, is the dividend, because it is the amount we must deliver. The concentration, 0.01 g/mL, is the divisor, because it is the amount in each single millilitre. Volume = 0.25 ÷ 0.01 g/mL.
Carry out the division by moving both decimal points two places to the right.
Reveal this step
0.25 ÷ 0.01 becomes 25 ÷ 1 = 25, so the volume to draw up is 25 mL. Moving both points the same distance leaves the quotient unchanged.
Check the answer against the label. Does 25 mL of a 1% solution really contain the 250 mg that was ordered?
Reveal the answer
25 mL at 0.01 g/mL gives 25 × 0.01 = 0.25 g, and 0.25 g is 250 mg. The answer is consistent with the order, and the check took one multiplication.

Answer: The 1% strength is 0.01 g/mL, and the volume to draw up is 25 mL, which contains exactly the 250 mg ordered.

Keep this habit: Convert the percent to a decimal of grams per millilitre before dividing, keep the dose required as the dividend, and multiply the finished volume back by the concentration to prove that it holds the ordered amount.

Tutorial 2 — dividing by 0.01 without moving the wrong number

A paediatric vitamin D supplement supplies 0.01 mg of vitamin D in each drop. The prescription is 0.05 mg once a day. Find the number of drops, and show exactly which number is the dividend and which is the divisor.

Rule used: To find how many unit doses are needed: number of doses = amount required ÷ amount supplied by one unit. The amount required is the dividend and the amount supplied by one unit is the divisor. Dividing by 0.01 is the same as multiplying by 100, so the answer is larger than the dividend.

Write the two quantities with their units before you divide anything, and say what you are looking for.
Reveal this step
Amount required = 0.05 mg. Amount supplied by one drop = 0.01 mg. We are looking for a number of drops, which is a count and carries no unit of mass.
Which of the two is the divisor? Explain what goes wrong if you swap the dividend and the divisor.
Reveal the answer
0.01 mg per drop is the divisor, because it is the amount that one unit supplies. Swapping them would give 0.01 ÷ 0.05 = 0.2 drop for a larger order, which is impossible: a bigger dose cannot need less than one drop.
Set out 0.05 ÷ 0.01 and move the decimal point of the divisor to the right until the divisor is a whole number.
Reveal this step
Moving the divisor’s point two places turns 0.01 into 1. The divisor is now the whole number 1, and the layout is 0.05 ÷ 0.01 → ? ÷ 1.
The divisor’s point has moved two places. What must you do to the dividend, and why is that fair?
Reveal the answer
Move the dividend’s point the same two places: 0.05 becomes 5. Multiplying the dividend and the divisor by the same number, here 100, leaves the quotient unchanged, so 0.05/0.01 = 5/1.
Do the division, then prove the answer by multiplying back by the amount in one drop.
Reveal this step
5 ÷ 1 = 5, so the dose is 5 drops. Check: 5 drops × 0.01 mg per drop = 0.05 mg, which is exactly the amount prescribed.
How would you write the calculation so that a colleague checking it cannot swap the two numbers?
Reveal the answer
Write the labels with the numbers: (0.05 mg required) ÷ (0.01 mg per drop) = 5 drops. The unit of the answer tells the checker that the arrangement is right, because only this one leaves a count of drops.

Answer: The dose is 5 drops, because 0.05 ÷ 0.01 = 5, and 5 drops × 0.01 mg per drop returns the prescribed 0.05 mg.

Keep this habit: Never do a decimal division without naming the dividend and the divisor first. Dividing by 0.01 makes the answer a hundred times larger, so the result should always be bigger than the amount you started with.

Now you try

Attempt each one on paper first. Write the formula, substitute the numbers, then check the unit of your answer.

Compare 0.7 with 0.68, then add 14.8 + 6.29 + 3.028 by writing the numbers in a column with the decimal points in one vertical line.
Hint
Give every number the same number of decimal places by adding zeros: 0.70 and 0.680. Then work one column at a time from the right, carrying when a column reaches ten.
Show the worked answer
Written as 0.70 and 0.680, the tenths column gives 7 against 6, so 0.7 is the greater number even though 0.68 uses more digits. For the addition, write 14.800, 6.290 and 3.028; the thousandths give 8, the hundredths give 0 + 9 + 2 = 11 so carry 1, the tenths give 8 + 2 + 0 + 1 = 11 so carry 1, and the units give 4 + 6 + 3 + 1 = 14. The sum is 24.118, with the point kept directly below the others.
Multiply 0.132 by 100 and divide 5.897 by 0.01, stating for each how many places the decimal point moves and in which direction.
Hint
Count the zeros in 100. For 0.01, remember that dividing by a hundredth is the same as multiplying by 100, so the point travels the same way.
Show the worked answer
0.132 × 100 = 13.2: the multiplier has two zeros, so the point moves two places right. 5.897 ÷ 0.01 = 589.7: dividing by 0.01 is multiplying by 100, so the point again moves two places right and the answer is much larger than the dividend, as it must be when the divisor is less than 1.
Round 4.374 to the nearest tenth and 0.268 to two decimal places; then write 7/8 as a decimal by dividing the numerator by the denominator.
Hint
Look only at the single digit immediately to the right of the place you want. For 7/8, extend the division until it stops or until you have three decimal places.
Show the worked answer
4.374 to the nearest tenth: the digit after the tenths is 7, so round up to 4.4. 0.268 to two decimal places: the digit after the hundredths is 8, so round up to 0.27. 7 ÷ 8 = 0.875, so 7/8 = 0.875, a value that sits between 0.8 and 0.9 on the number line.
What percent of 24 is 9? Then find 12½% of 240 mg, changing the percent to a decimal before you multiply.
Hint
Put the number that follows the word “of” in the denominator and the other number in the numerator. For 12½%, write the mixed number as 12.5 and then divide by 100.
Show the worked answer
9 ÷ 24 = 0.375, and 0.375 × 100 = 37.5, so 9 is 37.5% of 24. For the second part, 12½% = 12.5 ÷ 100 = 0.125; then 0.125 × 240 mg = 30 mg. The answer is about one eighth of the whole, which matches 12½% and confirms that the point was placed correctly.

Common misconception

Beginners often assume that the decimal with more digits after the point is the larger number, so 0.0096 looks bigger than 0.432 and 0.68 looks bigger than 0.7. The digits are not the value; the position of the point is, and adding a zero to the right of a decimal changes nothing at all — 0.7 and 0.70 are the same quantity. Standing the numbers in a column with the points aligned exposes this immediately, because the comparison is then made digit by digit at the same place value. The same mistaken model produces the opposite error when multiplying: learners expect a product to be larger than its factors, yet two decimals below 1 must produce a smaller answer, as 0.5 × 0.5 = 0.25 shows.

Check your reasoning

Explain in your own words why dividing by 0.01 makes a number larger, while dividing by 100 makes it smaller, even though 0.01 and 100 are related.

Reveal an answer

0.01 and 100 are reciprocals: 0.01 = 1/100. Dividing by a number is the same as multiplying by its reciprocal, so dividing by 0.01 is multiplying by 100, and dividing by 100 is multiplying by 0.01. The size of the divisor decides the direction of the change, not the word “division”. Whenever the divisor is smaller than 1, the quotient is larger than the dividend; whenever the divisor is greater than 1, the quotient is smaller. Checking the divisor against 1 before you calculate is therefore a free error test.

Where this comes from

BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (supplied) · Unit One, Lesson 2 “Decimals” (lines 177–260): decimal fractions and place value, the mandatory zero before the point, comparing decimals by column, addition and subtraction, multiplication, multiplying by 10 and by 0.1 multiples, rounding to a named place, division by 0.1 or a multiple of 0.1, and decimal–fraction conversion. Lesson 3 “Percents” (lines 261–299): percent as hundredths, changing a percent to a fraction and to a decimal including percents containing a fraction, finding what percent one number is of another, and finding a given percent of a number.
Note on the course code · The supplied cover page reads “Course Number: BMS 110”. The curriculum catalogue in this project records “BMS 1110”; the catalogue is treated as authoritative and the notes’ cover value is retained here for provenance.
Beyond the supplied notes: leading and trailing zeros as a medication-safety rule · The notes state the rule about writing a zero before the decimal point (line 185) but do not explain the mechanism or the matching prohibition on trailing zeros. The tenfold and hundredfold misreading mechanism, the never-a-trailing-zero convention and the label-reading habit are standard medication-safety teaching added here.
Beyond the supplied notes: base-ten exponents behind the moving point · The notes give the movement rules for multiplying and dividing by powers of ten and by 0.1 multiples (lines 223–249) as procedures. The exponent explanation (10⁻¹, 10⁻², 10⁻³), the reciprocal argument that dividing by 0.01 is multiplying by 100, and the link to metric-prefix conversions are standard place-value theory supplied here.
Beyond the supplied notes: percentage points, percent change and percent strength as concentration · The notes do not distinguish percentage points from relative change or name percentage strength as a concentration with a unit carried by its denominator of 100. Both are standard clinical-numeracy content added in the deeper topics.

Lesson progress is saved on this device for the current learner. Practice answers stay separate and are not stored.

About this course and its sources

Bilingual lessons organised from the supplied BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (Mr. Ebangan Prosper Achuanoh). Diagrams are original educational schematics; sections that go beyond the supplied notes say so in their sources. Mathematics, clinical and French-language review are pending.