Another Peek Beyond the Point | CBSE Class 7 Maths Notes

This note covers decimal place value, multiplication and division of decimals, fractions and decimal quotients, the size of products and quotients, repeating decimal digits, applications to measurement and money, leap-year calculations, and the Hidato number puzzle.

How does place value explain a decimal number?

A decimal represents a number using place values that extend beyond ones to parts of a whole. A decimal fraction has a denominator such as 10, 100 or 1000. The denominator is the number below the fraction bar; the numerator is the number above it.

The decimal point separates the ones place from the tenths place. A tenth is one of ten equal parts of a whole, a hundredth is one of a hundred equal parts, and a thousandth is one of a thousand equal parts.

In the notation used here, = means “is equal to”, + means addition, − means subtraction, × means multiplication, and ÷ means division. A slash in a fraction, such as 1/10, separates its numerator and denominator.

How can we read the value of each digit?

In 27.53, the 2 represents two tens, the 7 represents seven ones, the 5 represents five tenths and the 3 represents three hundredths. Reading the places explains the quantity represented by the digits on both sides of the point.

The fraction 254/1000 separates into 200/1000 + 50/1000 + 4/1000. These parts are 2/10, 5/100 and 4/1000. In decimal form, 0.254 = 0.2 + 0.05 + 0.004.

This is the connection between fractions and decimals: the digits record quantities of tenths, hundredths and smaller parts. Multiplication and division of decimals build on the same quantities, so place value explains why the calculation procedures work.

How do we divide by 10, 100 and 1000?

Definition: In division, the dividend is the number being divided, the divisor is the number by which it is divided, and the quotient is the result.

Property: Dividing by a power of ten

A power of ten, as used here, is a number such as 10, 100 or 1000, written as 1 followed by zeroes. Dividing by one of these numbers moves the decimal point left by as many places as there are zeroes in the divisor.

For a counting number, meaning a number in the sequence 1, 2, 3 and so on, begin with the decimal point after its last digit. Add zeroes before the digits when needed to show the new place values.

Worked example 1. Find 123 ÷ 10 and 12 ÷ 1000.

Answer: For 123 ÷ 10, start with 123. and move the point one place left to get 12.3. For 12 ÷ 1000, move it three places left, supplying the zero needed between the point and 1. The answer is 0.012.

How does the rule work with a decimal dividend?

The same rule applies when the dividend already has a decimal point. For example, 18.7 ÷ 10 = 1.87, 18.7 ÷ 100 = 0.187, 18.7 ÷ 1000 = 0.0187 and 18.7 ÷ 10000 = 0.00187.

Anuja divides a ribbon measuring 3.9 metres into equal pieces. The abbreviation m means metre, a unit of length. Ten equal pieces each measure 3.9 ÷ 10 = 0.39 m. A hundred equal pieces each measure 3.9 ÷ 100 = 0.039 m.

The fraction calculation explains the second result: 3.9 = 39/10, and dividing this by 100 gives 39/1000. The digits retain their order, but their place values become smaller. Count the movement carefully instead of simply removing or adding a zero.

How does decimal multiplication arise from fractions?

Multiplication can describe repeated addition when we take a whole number of equal quantities. The product is its result. The numbers multiplied are factors, also described as the multiplier and multiplicand. Converting them into fractions extends the method to two decimal quantities.

How can a purchase be calculated in two ways?

Worked example 2. Arshad buys 5 pens at ₹9.5 each. Find the total cost. The symbol ₹ means rupees.

Answer: Add 9.5 five times to obtain 47.5. Alternatively, write 9.5 as 95/10 and 5 as 5/1. Then (95/10) × (5/1) = 475/10 = 47.5. The five pens cost ₹47.5.

To multiply fractions, multiply their numerators together and their denominators together. In the pen calculation, the denominator remains 10 because 10 × 1 = 10. The answer therefore represents 475 tenths, which is the same quantity as 47.5.

How does a decimal quantity multiply another decimal?

Worked example 3. A car travels 12.5 kilometres per litre of petrol. Find the distance covered using 7.5 litres. A kilometre, abbreviated km, is a unit of length; a litre is a unit of capacity.

Answer: Distance covered = 12.5 × 7.5 = (125/10) × (75/10) = 9375/100 = 93.75 km. The product has two decimal places because the denominator is 100.

Here, per litre means for each litre. Both quantities have one decimal place, but their product is expressed in hundredths. The denominator comes from 10 × 10, so the two original decimal places both matter.

These calculations show why a decimal point cannot be placed by guesswork. First identify the operation from the situation. Then use fractions or the decimal multiplication rule to retain the correct value of the answer.

How do we place the decimal point in a product?

Property: Decimal places in multiplication

A decimal place is a position to the right of the decimal point. Multiply the numbers as counting numbers after removing their decimal points. Then count the total decimal places in the original factors and place the point that many digits from the right in the product.

The reason is the multiplication of the denominators. A denominator of 10 multiplied by 100 gives 1000. There is one zero in 10 and two in 100, so the combined denominator has three zeroes and gives three decimal places.

Worked example 4. Given 596 × 248 = 147808, calculate 5.96 × 24.8.

Answer: The first factor has two decimal places and the second has one. Their total is three. Place the decimal point three positions from the right in 147808 to obtain 147.808. Thus 5.96 × 24.8 = 147.808.

How can the rule be checked?

Worked example 5. Find the product of 5.8 and 1.24.

Answer: Multiply 58 × 124 = 7192. The factors have one and two decimal places, so their product is 7.192. The fraction check is (58/10) × (124/100) = 7192/1000 = 7.192.

Do not use the procedure for adding decimals here. In multiplication, it is the sum of the decimal-place counts that determines the point. Keeping track of denominators explains the count and provides a second way to check the result.

The calculations 1.64 × 6 = 9.84 and 5.7 × 13.35 = 76.095 illustrate two further arrangements. In the first, the counting-number factor contributes no decimal places. In the second, the factors contribute one and two decimal places respectively.

How are decimal products used in length and area?

When quantities describe measurements, the answer must include the appropriate unit, the agreed measure used to express a quantity. A metre and a kilometre both measure length. For these calculations, 1 km = 1000 m, so metres can be converted into kilometres by dividing by 1000.

How do we account for repeated journeys?

Worked example 6. Ajay lives 827 m from school. He walks to school and back each day for 6 days. Find his weekly walking distance in kilometres.

Answer: Each journey is 0.827 km. Going to school and returning gives 0.827 × 2 = 1.654 km per day. Over 6 days he walks 1.654 × 6 = 9.924 km.

The factor 2 accounts for the two journeys each day. The factor 6 accounts for the number of days. Keeping these two steps separate makes it clear that 827 m is the one-way distance, rather than the total distance for a day.

How is the area of a rectangle calculated?

A rectangle is a four-sided figure with four right angles, each equal to a quarter turn. Its area measures the surface it covers. Multiply its length by its breadth, the lengths of two adjacent sides. The abbreviation cm means centimetre, and cm² means square centimetre, a unit of area.

The horizontal rectangle has its left side labelled 5.7 cm and its bottom side labelled 13.3 cm.

Reference: NCERT Class 7, page 70, unnumbered figure

Worked example 7. Find the area of a rectangle with adjacent sides measuring 5.7 cm and 13.3 cm.

Answer: Area = 5.7 × 13.3 = (57/10) × (133/10) = 7581/100 = 75.81 cm². There are two decimal places in the product because each side measurement has one decimal place.

Is a decimal product greater than both factors?

A multiplication does not necessarily produce an answer larger than both numbers being multiplied. To predict the size of a product of positive numbers, meaning numbers greater than zero, compare each factor with 1 before doing the full calculation.

Property: Product size depends on the factors

If both factors are greater than 1, their product is greater than both. If both are between 0 and 1, their product is less than both. With one factor in each range, the product lies between the two factors.

How do contrasting examples help?

Compare 2.25 × 8 = 18 with 0.25 × 8 = 2. In the first calculation, 18 exceeds both factors. In the second, 2 exceeds 0.25 but is below 8. The decimal point changes the size of the factor and therefore the relationship.

Now compare 0.25 × 0.8 = 0.2. Both factors lie between 0 and 1, and the product is smaller than either. These examples also show that a product involving decimals can be a counting number, as in 0.25 × 8 = 2.

Use size comparisons as a check on the decimal point. They do not replace the calculation, but they help reveal an answer in the wrong range. A result should agree with both the digit calculation and the size expected from its factors.

How can equivalent fractions give decimal quotients?

Equivalent fractions express the same quantity using different numerators and denominators. Multiplying both parts of a fraction by the same non-zero number preserves its value. This helps turn a division into a decimal when the denominator can become 10, 100 or 1000.

How can a ribbon be shared equally?

Neenu has 29 m of ribbon to share equally with Anu. Dividing 29 by 2 gives each person 14 m with 1 m left. Half of that remaining metre is 1/2 = 5/10 = 0.5 m. Each person therefore receives 14.5 m.

Worked example 8. Find the length each friend receives when 29 m of ribbon is shared equally among four friends.

Answer: Each share is 29/4 m. Since 4 × 25 = 100, multiply the numerator and denominator by 25: (29 × 25)/(4 × 25) = 725/100 = 7.25. Each friend receives 7.25 m.

The choice of 25 is purposeful: it converts the denominator 4 into 100. Multiplying the numerator as well prevents the quantity from changing. Once the fraction has denominator 100, its value is read in hundredths.

Why do we also need long division?

The fraction 10/3 cannot be changed into an equivalent fraction with denominator 10, 100, 1000 or another such power of ten. A more general method is therefore needed. Long division divides a number place by place and regroups any amount left over.

A remainder is the amount left at a division step. Regrouping rewrites that amount in smaller place-value units without changing its value. One remaining one becomes ten tenths, allowing the division to continue beyond the ones place.

How does long division continue beyond the ones place?

Long division starts at the larger place values and works towards the smaller ones. It does not need to stop when a remainder is left after the ones. The remainder can be regrouped into tenths, then hundredths, then thousandths as required.

What happens when 1325 is divided by 4?

Worked example 9. Calculate 1325 ÷ 4 by place value.

Answer: Each share receives 3 hundreds, 3 tens and 1 one, with 1 one remaining. Regroup that one as 10 tenths. Each share receives 2 tenths, leaving 2 tenths. Regroup these as 20 hundredths and give each share 5 hundredths. The quotient is 331.25.

The decimal point belongs between the ones and tenths in the quotient. It is inserted when the division moves from sharing ones to sharing tenths. Each later digit must keep its place, because tenths and hundredths represent different quantities.

Arrows distribute 20 hundredths among four boxes. The completed boxes each contain 3 hundreds, 3 tens, 1 one, 2 tenths and 5 hundredths.

Reference: NCERT Class 7, page 80, unnumbered figure

How can a quotient need three decimal places?

For 237 ÷ 8, the whole-number part is 29 and the remaining amount is 5 ones. The successive regroupings explain each decimal digit:

1. Regroup the 5 ones as 50 tenths.
2. Divide 50 tenths by 8 to give 6 tenths per share, leaving 2 tenths.
3. Regroup these as 20 hundredths; each share receives 2 hundredths, leaving 4 hundredths.
4. Regroup these as 40 thousandths; each share receives 5 thousandths, with no remainder.

The quotient is 29.625. The digits after the point record six tenths, two hundredths and five thousandths. Continuing until no remainder remains gives the complete value in this calculation.

How do we divide when the dividend is already a decimal?

A decimal dividend is divided using the same place-value method. First share the ones. Then place the decimal point in the quotient before sharing the tenths. Any remaining amount is combined with the quantity already present in the next smaller place.

How is sugar shared between four bags?

Worked example 10. A shopkeeper divides 9.5 kilograms of sugar equally among 4 bags. Find the amount in each bag. The abbreviation kg means kilogram, a unit of mass.

Answer: Sharing 9 ones gives 2 ones per bag and leaves 1 one. Regroup it into 10 tenths and combine it with the existing 5 tenths. Divide the resulting 15 tenths, then continue through hundredths and thousandths. The quotient is 2.375 kg per bag.

At the tenths step, 15 ÷ 4 gives 3 tenths per bag and leaves 3 tenths. These become 30 hundredths. Sharing gives 7 hundredths per bag and leaves 2 hundredths. They become 20 thousandths, giving 5 thousandths per bag.

Why are zeroes in the quotient necessary?

Worked example 11. Find 0.06 ÷ 5.

Answer: There are no ones or tenths to share, so the quotient begins 0.0. Six hundredths divided by 5 gives 1 hundredth and leaves 1 hundredth. Regroup that remainder as 10 thousandths. Each share receives 2 thousandths, making the quotient 0.012.

The zero after the decimal point records zero tenths. Leaving it out would move the other digits into the wrong places. Reading the answer as one hundredth and two thousandths explains both the zero and the positions of 1 and 2.

How do we divide by a decimal divisor?

A decimal divisor can be changed into a counting number before using long division. Multiply it by a suitable power of ten. Multiply the dividend by exactly the same number so that the quotient remains unchanged.

Property: Scaling both numbers preserves a quotient

To calculate 4.68 ÷ 1.3, multiply both numbers by 10 to obtain 46.8 ÷ 13. For 4.68 ÷ 0.13, multiply both by 100 to obtain 468 ÷ 13. The required multiplier is determined by the divisor.

Worked example 12. Ravi travels 126 km from Pune to Matheran in 2.5 hours. Find his average speed, meaning total distance divided by total time.

Answer: Average speed = 126 ÷ 2.5. Multiplying both numbers by 10 gives 1260 ÷ 25 = 50.4. Ravi's average speed is 50.4 km per hour, meaning kilometres travelled per hour on average over the journey.

Does division necessarily make the answer smaller?

Compare 128 ÷ 4 = 32 and 128 ÷ 0.4 = 320. The first quotient is less than its dividend; the second is greater. Therefore, the word “division” alone does not tell us whether an answer must be smaller.

For a positive dividend, division by a number greater than 1 gives a smaller quotient. Division by a positive number less than 1 gives a larger quotient. Dividing by 1 leaves the dividend unchanged. These conditions matter when checking a result.

In a decimal-divisor calculation, check the two changes together. Moving the point in only the divisor changes the problem. After making the divisor a counting number, use place-value division and check whether the result has the expected size.

Why do some decimal divisions never end?

Some long divisions continue because a remainder keeps appearing. A finite decimal has a limited number of digits after the decimal point. A repeating decimal continues with a digit or block of digits recurring in the same order.

Why does 10 divided by 3 keep producing threes?

In 10 ÷ 3, sharing ten ones gives 3 ones with 1 one left. Regrouping this as ten tenths gives 3 tenths with 1 tenth left. Sharing ten hundredths next gives 3 hundredths with 1 hundredth left.

The same remainder pattern keeps returning, so this division never ends. We write 10 ÷ 3 = 3.333.... The three dots, called an ellipsis, indicate that the digits continue. This quotient cannot be expressed using a finite number of decimal digits.

What repeats when 1 is divided by 7?

For 1 ÷ 7, the remainders cycle through 1, 3, 2, 6, 4 and 5 before returning to 1. The quotient digits repeat as 142857, giving 0.142857142857... . Both the remainder cycle and the digit cycle continue.

Arrows link 1, 3, 2, 6, 4 and 5, then return to 1 and continue through the same sequence.

Reference: NCERT Class 7, page 85, unnumbered figure

The block 142857 also gives an interesting multiplication pattern. Multiplying it by counting numbers from 1 to 6 produces the same digits cycled around. The repeating block in 1 ÷ 17 provides another number to investigate in this way.

A recurring remainder explains why continuing long division does not necessarily lead to a final digit. Recognising the return to an earlier remainder helps identify the repeated part without treating a shortened decimal as the complete answer.

How do decimals explain leap years and calendar adjustments?

The calendar calculation uses 365.2422 days for one revolution of Earth around the Sun. A revolution means one complete journey around the Sun. A year of 365 days leaves 0.2422 days unaccounted for; over 100 such years this becomes 24.22 days.

The Gregorian calendar uses ordinary years and leap years, with the extra days determined by the divisibility rules below. A leap year contains 366 days rather than 365. Decimal multiplication helps compare successive arrangements for adding these extra days and shows why the first adjustment needs further changes.

Why is one extra day every four years insufficient?

Adding a day every fourth year gives 4 × 365 + 1 = 1461 calendar days. The corresponding calculation for Earth's revolutions is 4 × 365.2422 = 1460.9688 days. The calendar has slightly more days in this comparison.

Over 100 years, this first arrangement gives 36,525 days, compared with 36,524.22 days for the revolutions. Omitting the extra day in the hundredth year instead gives 24 leap years and 76 other years: (24 × 366) + (76 × 365) = 36,524 days.

Repeating the adjusted hundred-year total ten times gives 3,65,240 calendar days. The calculation for 1000 revolutions gives 3,65,242.2 days, leaving a difference of 2.2 days. Restoring an extra day every 400th year provides the next adjustment.

What is the final leap-year rule?

Divisible means division leaves no remainder. A year divisible by 400 has 366 days. Otherwise, a year divisible by 100 has 365 days. Among the remaining years, those divisible by 4 have 366 days and the others have 365 days.

The flowchart tests divisibility by 400 first, then by 100, then by 4. Its Yes and No branches lead to a year of either 366 or 365 days.

Reference: NCERT Class 7, page 91, unnumbered figure

For years 1 through 1000, the final calculation is (750 × 365) + (240 × 366) + (8 × 365) + (2 × 366) = 3,65,242 days. The revolution calculation gives 3,65,242.2 days. The calendar is slightly shorter, by 0.2 days over these 1000 years.

The successive adjustments illustrate how a small decimal difference can accumulate. The final arrangement retains a small difference; it does not make these two totals exactly equal.

How can we approach the applications and number puzzle?

Begin an application by identifying the quantity wanted and the unit required. Equal shares suggest division; repeated quantities suggest multiplication. Read whether a distance is one-way or includes a return journey, and whether a price is for each item or for a measured quantity.

How do conversions fit into a calculation?

The unit relationships include 1 cm = 10 millimetres, 1 m = 100 cm, 1 km = 1000 m, 1 kg = 1000 grams, 1 gram = 1000 milligrams and 1 litre = 1000 millilitres. These relate larger units to smaller units of the same kind.

For example, the cloth problem asks for three shirts, each needing 1.65 m. Multiplication gives the total. The juice problem shares 3 litres equally among 8 friends and asks for millilitres, so equal sharing and unit conversion must both be addressed.

The bookshelf problem gives books 2.5 cm thick and a shelf 160 cm long. Division determines how many books fit. Such questions require interpreting the calculated quotient in the situation, rather than stopping with an unexplained number.

What is the Hidato challenge?

Hidato is a puzzle in which numbers are placed in a grid to make one continuous path. Consecutive numbers follow one another in counting order. Adjacent cells are neighbouring cells; this puzzle also allows diagonally adjacent cells, which meet at a corner.

The grid is usually square-shaped, but other shapes can be used. It can have inner holes while remaining one piece. Usually, the lowest and highest numbers are supplied, together with some intermediate numbers. The supplied numbers are chosen so that the puzzle has a single solution.

Fill the remaining cells so that each next number is adjacent to the previous number, including diagonally. This uses a different kind of checking from decimal arithmetic: each entry must fit the continuous path and the numbers already supplied in the grid.

Glossary

• Decimal — A number representation using place values that extend from ones to tenths, hundredths and smaller parts.
• Decimal fraction — A fraction whose denominator is a power of ten, such as ten, hundred or thousand.
• Numerator — The number above a fraction bar, indicating how many of the fractional parts are taken.
• Denominator — The number below a fraction bar, indicating the equal parts into which a whole is divided.
• Decimal place — A digit position to the right of the decimal point, such as tenths or hundredths.
• Product — The result obtained when two or more given numbers are multiplied together.
• Dividend — The number being divided into shares or groups in a division calculation.
• Divisor — The number by which the dividend is divided to obtain the quotient.
• Quotient — The result of division, which may contain a finite or continuing decimal part.
• Remainder — The amount left over at a particular step after the available quantity is divided.
• Equivalent fractions — Fractions with different numerators and denominators that represent the same numerical quantity.
• Regrouping — Rewriting a quantity in smaller place-value units so that division can continue.
• Repeating decimal — A decimal in which a digit or block of digits continues to recur.
• Leap year — A calendar year containing three hundred and sixty-six days instead of three hundred and sixty-five.

Common errors and misconceptions

• Misconception: Every product is larger than both its factors. Correct: When both factors are between 0 and 1, their product is smaller than both.
• Misconception: Decimal multiplication uses only the larger decimal-place count. Correct: Add the counts from both factors when positioning the decimal point in their product.
• Misconception: Division stops when there is a remainder after the ones place. Correct: Regroup the remaining ones as tenths and continue into smaller place values.
• Misconception: The zero after the point in 0.012 can be omitted. Correct: It records zero tenths; omitting it changes the value of the number.
• Misconception: To divide by a decimal, make only the divisor a counting number. Correct: Multiply the dividend and divisor by the same power of ten.
• Misconception: Division by a decimal gives a quotient greater than the dividend in every case. Correct: For positive numbers, the effect depends on whether the divisor is below, equal to or above 1.
• Misconception: Every decimal quotient eventually ends. Correct: Some divisions repeat a remainder pattern, causing the decimal digits to continue indefinitely.
• Misconception: Every year divisible by 4 is a leap year without further checks. Correct: Years divisible by 100 require the additional divisibility-by-400 test.

Exam-style questions with model answers

Q1. Calculate 5.8 × 1.24 and explain the decimal-point position. [2 marks]

1. Removing the decimal points gives 58 × 124 = 7192.
2. The factors have one and two decimal places, giving three altogether. Therefore, the product is 7.192.

Q2. Anuja cuts a 3.9 m ribbon into 100 equal pieces. Find the length of each piece in metres and explain your calculation. [2 marks]

1. The required equal share is 3.9 ÷ 100, because the total length is divided among 100 pieces.
2. Move the decimal point two places left. Each piece measures 0.039 m.

Q3. Ajay lives 827 m from school and walks to school and back each day for 6 days. Using 1 km = 1000 m, find his total walking distance in kilometres. [3 marks]

1. Convert the one-way distance into kilometres: 827 ÷ 1000 = 0.827 km. This is the distance between home and school.
2. He makes two journeys each day, so the daily walking distance is 0.827 × 2 = 1.654 km.
3. For six days, multiply the daily total by 6: 1.654 × 6 = 9.924 km.

Q4. Find 0.06 ÷ 5 using place value. Explain the zeroes and the regrouping in your answer. [4 marks]

1. There are zero ones in the dividend. Sharing them among five parts gives zero ones in the quotient.
2. Place the decimal point before the tenths digit. There are also zero tenths, so the quotient begins 0.0.
3. Six hundredths divided by five gives one hundredth per share, leaving one hundredth to be divided further.
4. Regroup the remaining hundredth as ten thousandths. Each share receives two thousandths, so the complete quotient is 0.012.

Q5. Ravi travels 126 km in 2.5 hours. Average speed is total distance divided by total time. Calculate his average speed in kilometres per hour, showing how to handle the decimal divisor. [3 marks]

1. Use the given definition of average speed: divide the total distance of 126 km by the total time of 2.5 hours.
2. Multiply both the dividend and the divisor by 10. The quotient is unchanged, giving the equivalent calculation 1260 ÷ 25.
3. Dividing gives 50.4. Ravi's average speed over the journey is therefore 50.4 kilometres per hour.

Q6. Use long division to explain why 10 ÷ 3 has no final decimal digit, and write the quotient using dots to show continuation. [5 marks]

1. Divide ten ones into three equal shares. Each share receives three ones, with one one left over to divide at the next place.
2. Regroup that remaining one as ten tenths. Each share receives three tenths, and one tenth remains. Place the decimal point before the tenths digit.
3. Regroup the remaining tenth as ten hundredths. Dividing by three gives three hundredths in each share and leaves one hundredth.
4. At each new place, the same pattern returns: ten smaller units are divided by three, leaving one smaller unit. The remainder therefore does not disappear.
5. The quotient is 3.333... . The dots indicate continuing threes. A finite number of decimal digits cannot give the complete result of this division.

Q7. Use 365.2422 days for each revolution of Earth. In years 1 through 1000, suppose 750 years and another 8 years have 365 days, while 240 years and another 2 years have 366 days. Calculate both day totals and their difference, and state which is shorter. [5 marks]

1. The first group contributes 750 × 365 = 2,73,750 days. The second group with the same year length contributes 8 × 365 = 2,920 days.
2. The groups with 366 days contribute 240 × 366 = 87,840 days and 2 × 366 = 732 days respectively.
3. Adding the four contributions gives 2,73,750 + 2,920 + 87,840 + 732 = 3,65,242 calendar days over the specified thousand years.
4. Using the supplied duration for a revolution, the corresponding total for one thousand revolutions is 1000 × 365.2422 = 3,65,242.2 days.
5. The difference is 3,65,242.2 − 3,65,242 = 0.2 days. The calendar total is slightly shorter; these totals are close but not exactly equal.

Key takeaways

• Decimal digits represent tenths, hundredths and smaller parts, extending the same place-value structure used for counting numbers.
• Multiply decimals as counting numbers first, then use the total decimal-place count from both original factors.
• A product can be smaller than both factors when each factor lies between zero and one.
• Long division continues beyond ones by regrouping remainders into tenths, hundredths and smaller place-value units.
• To remove a decimal divisor, multiply both divisor and dividend by the same suitable power of ten.
• Some divisions never end because repeating remainder patterns produce a repeating digit or block of decimal digits.
• Read the required unit carefully, since measurement problems may need conversion as well as multiplication or division.
• Leap-year calculations show how small decimal differences accumulate and why successive calendar adjustments are needed.

Test yourself

What does the digit 3 represent in 27.53?

It represents three hundredths, because it is in the second place after the decimal point.

What is 12 ÷ 1000?

The quotient is 0.012, obtained by moving the decimal point three places left.

Given 596 × 248 = 147808, what is 5.96 × 24.8?

The product is 147.808 because the two factors contribute three decimal places altogether.

Why is 0.75 × 0.4 smaller than both factors?

Both factors lie between zero and one, so their product, 0.3, is smaller than each.

When is the decimal point inserted during place-value division?

Insert it in the quotient when moving from the ones place to the tenths place.

How can 4.68 ÷ 0.13 be rewritten with a counting-number divisor?

Multiply both numbers by 100 to obtain the equivalent calculation 468 ÷ 13.

Which block of digits repeats in the decimal quotient of 1 ÷ 7?

The block 142857 repeats, giving the continuing decimal representation 0.142857142857... .

What extra condition makes a year divisible by 100 a leap year?

It must also be divisible by 400 to have 366 days under the final rule.