Another Peek Beyond the Point
Go beyond reading decimals — multiply them, divide them, discover repeating patterns, and use decimals to understand the calendar.
From coins and ribbons to cyclic numbers and leap years, explore how decimal operations reveal patterns hiding beyond the decimal point. Multiply decimals as fractions, trace long-division regrouping into Tenths and Hundredths, meet the famous cyclic number 142857, and discover why Earth's 365.2422-day solar orbit requires the 400-year leap-year rule!
What You Will Discover in This Chapter
4.1 A Quick Recap of Decimals
Indian Place-Value System, Market Weights, Fraction Expansions & Powers of 10
Interactive Place-Value Chart (27.53)
Decimals extend the Indian place-value system to represent fractions with denominators 10, 100, 1000, etc. Each place to the right is one-tenth of the previous place. Adjust the sliders below to see the place-value decomposition:
Jonali's Spice Market Purchases (Grams to Kilograms)
1 kilogram = 1000 grams. Each gram is 1/1000 kg (0.001 kg):
Powers of 10: The Decimal Shift Rule
Dividing by 10, 100, or 1000 shifts the decimal point left. Each digit moves to a smaller place value (ones become tenths, tenths become hundredths, etc.). This is the key insight behind writing market weights as decimals!
Every decimal can be decomposed into a sum of fractions with denominators 10, 100, 1000, …
4.2 Decimal Multiplication
Pen Shop, Petrol Mileage, Walking Distances, and the Decimal Place Counting Rule
Arshad's Stationery Shop (5 Pens at ₹9.50 each)
Arshad buys 5 pens. If one pen costs ₹9.50, what is the total cost?
9.5 + 9.5 + 9.5 + 9.5 + 9.5 = ₹47.50
(95/10) × (5/1) = (95 × 5) / 10 = 475 / 10 = ₹47.50
Notice: 1 decimal place + 1 decimal place = 2 decimal places in the product!
Is the Product Always Greater than the Numbers Multiplied?
3.4 × 6.5 = 22.1
Product is GREATER than both numbers.0.75 × 0.4 = 0.30
Product is LESS than both numbers!0.75 × 5 = 3.75
Product is in between the two numbers.Ajay's Daily Walking Distance
Ajay walks 1.35 km every day. Drag the slider to find his total distance over any number of days:
Rectangle Area with Decimal Dimensions
Adjust the length and width of a rectangle. Notice the decimal place counting rule in action:
Product Pattern Lab: Decimal Places Stack!
Start with 18 × 12 = 216. The digits stay the same but the decimal point shifts as you add decimal places to the factors:
| Expression | Decimal Places | Product |
|---|---|---|
| 18 × 12 | 0 + 0 = 0 | 216 |
| 18 × 1.2 | 0 + 1 = 1 | 21.6 |
| 1.8 × 12 | 1 + 0 = 1 | 21.6 |
| 1.8 × 1.2 | 1 + 1 = 2 | 2.16 |
| 0.18 × 0.12 | 2 + 2 = 4 | 0.0216 |
Figure It Out — Page 73
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4.3 Decimal Division
Ribbon Cutting, Long Division with Regrouping, Decimal Divisors, and Cyclic Number 142857
Anuja's Ribbon Cutting (3.9 m Ribbon)
Anuja cuts 3.9 m of ribbon into equal pieces. Choose the number of pieces to see how the decimal point shifts:
Long Division with Place-Value Regrouping: 1325 ÷ 4 = 331.25
(4 × 331 = 1324; remainder = 1).
Meet the Magic Cyclic Number: 142857 (from 1 ÷ 7)
When 1 is divided by 7, the remainders cycle endlessly: 1 → 3 → 2 → 6 → 4 → 5 → 1. The resulting repeating block of digits is 142857. Choose a multiplier from 1 to 7 below:
1 ÷ 17 produces a 16-digit cyclic block: 0588235294117647. In 1927, Austrian mathematician Emil Artin conjectured that there are infinitely many cyclic numbers arising from prime reciprocals. Nearly a century later, this remains an unsolved problem!
Divisor Slider Lab: How Divisor Affects the Quotient (Fixed Dividend = 12)
Neenu's Ribbon: Division via Fractions (29 ÷ 2 and 29 ÷ 4)
Neenu has 29 m of ribbon to share equally. What is each person's share? We convert the division to a fraction, then expand to decimal form:
Decimal Divisor Lab: Convert, Then Divide!
When the divisor is a decimal, multiply both dividend and divisor by the same power of 10 to make the divisor a whole number. Try the textbook examples:
Repeating Decimal Explorer
Some divisions never terminate — the remainder cycles! Enter a numerator and denominator (numerator less than denominator) to see the division unfold:
Remainder Cycle Visual: 1 ÷ 7 = 0.̄142857̄
When dividing 1 by 7, the remainders cycle in a fixed order forever: 1 → 3 → 2 → 6 → 4 → 5 → 1 → ...
| Step | Dividend (rem × 10) | ÷ 7 | Quotient digit | Remainder |
|---|---|---|---|---|
| 1 | 10 ÷ 7 | = 1 rem 3 | 1 | 3 |
| 2 | 30 ÷ 7 | = 4 rem 2 | 4 | 2 |
| 3 | 20 ÷ 7 | = 2 rem 6 | 2 | 6 |
| 4 | 60 ÷ 7 | = 8 rem 4 | 8 | 4 |
| 5 | 40 ÷ 7 | = 5 rem 5 | 5 | 5 |
| 6 | 50 ÷ 7 | = 7 rem 1 | 7 | 1 |
Figure It Out — Page 83: Decimal Division
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4.4 Look Before You Leap!
Earth's 365.2422-Day Solar Orbit, Calendar Drift & The Complete 400-Year Leap Year Rule
Earth's Solar Orbit vs 365-Day Calendar Drift
Earth takes 365.2422 days to orbit the Sun, but our calendar year is 365 days. That leaves an extra 0.2422 days every year. In 100 years, this accumulates to 0.2422 × 100 = 24.22 days!
Add 1 day every 4th year. In 100 years: 25 leap days = 36,525 days.
Actual solar days = 36,524.22 days.
Do not add an extra day in century years (divisible by 100).
Now 24 leap years per century = 36,524 days.
Every 400th year IS a leap year! (e.g. 1600, 2000, 2400).
Calendar days in 1000 years = 365,242 days.
Figure It Out — Pages 93–95: Multiplication & Division Challenges
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The Textbook Hidato Number Path Puzzle (1 to 40)
Fill the grid to create a continuous unbroken path from 1 to 40 moving horizontally, vertically, or diagonally!
Practice Zone (40 Graded Questions)
Progress through Foundation, Application, Challenge, and Master tiers
In the number 27.53, what is the place value of the digit 5?
What is 123 ÷ 10?
What is 24 ÷ 100?
If 1 pen costs ₹9.50, what is the cost of 5 pens?
How many decimal places will the product of 4.23 and 3.7 have?
Anuja cuts a 3.9 m ribbon into 10 equal pieces. What is the length of each piece?
Share a 29 m ribbon equally between 2 girls. What is each girl's share in decimal form?
To calculate 4.68 ÷ 1.3, what equivalent whole-number divisor problem should we solve?
How long does the Earth take to complete one full revolution around the Sun?
Is the year 2024 a leap year?
Ajay walks 827 m to school and 827 m back home, 6 days a week. How many kilometres does he walk in a week?
A car travels 12.5 km per litre of petrol. How far will it travel on 7.5 litres?
Find the area of a rectangle with length 13.3 cm and width 5.7 cm.
Given that 596 × 248 = 147808, what is 5.96 × 24.8?
Which of the following products is strictly LESS than 1 without calculating?
What is the decimal quotient of 1325 ÷ 4?
Ravi travels 126 km by scooter in 2.5 hours. What is his average speed?
A shopkeeper packs 9.5 kg of sugar equally into 4 bags. What is the weight of each bag?
Why was the year 1900 NOT a leap year even though it is divisible by 4?
Why WAS the year 2000 a leap year?
What is the decimal expansion of 10 ÷ 3?
What is the remainder cycle when 1 is divided by 7?
What remarkable result do you get when you multiply 142857 by 7?
When 128 is divided by 0.4, the quotient is 320. Why is the quotient greater than the dividend?
Without leap years, how many days would the calendar drift after 100 years?
Under the full Gregorian leap year rule (divisible by 4, except century years unless divisible by 400), what is the difference between calendar days and solar days in 1000 years?
On a number line from 3.1 to 3.2 divided into 10 equal parts, what number is located at the 6th mark?
On a number line from 2.15 to 2.17 divided into 10 equal parts, what number is located at the 6th mark?
In Sridharacharya's Patiganita problem, convert 6 1/4 ÷ 2 1/2 into decimals and solve.
Why do fractions with denominators like 2, 4, 8, 5, 25 always have terminating decimal expansions?
What did Austrian mathematician Emil Artin conjecture in 1927 about cyclic numbers like 142857?
How many digits are in the repeating decimal block of 1 ÷ 17?
Using digits 2, 4, 5, 8, and 0 exactly once, what is the MAXIMUM product of the form □□.□ × □.□?
Sort the following in increasing order: (a) 245.05 × 0.94, (b) 245.05 × 7.97, (c) 245.05 ÷ 7.97, (d) 245.05 ÷ 0.94, (e) 245.05, (f) 7.97.
A teacher wants to place 80 textbooks that are 2.5 cm thick on a 160 cm shelf. How many books can fit, and how much space is left?
210 g peanut chikki costs ₹70.50, while 110 g potato chips costs ₹33.25. Which item is cheaper per gram?
Shyamala buys 3 kg of bananas at ₹30/kg (counting 35 bananas in all) and sells them at ₹5 per banana. What is her profit?
In the division matrix a ÷ b where a = 15.17 and b = 0.37, what is the quotient?
In a Hidato puzzle, which adjacent cells can consecutive numbers connect through?
Over 10,000 years with the Gregorian calendar rule (97 leap years every 400 years = 2425 leap days), how many calendar days elapse compared to actual solar days?
Final Chapter Assessment (30 MCQs)
Evaluate your mastery across decimal multiplication, division, cyclic numbers, and leap years
Key Terms & Definitions (Decimals & Calendars)
A number expressed in the base-10 system using a decimal point to represent fractional parts based on tenths, hundredths, thousandths, etc.
The first place value to the right of the decimal point, representing 1/10 (0.1).
1/10 = 0.1The second place value to the right of the decimal point, representing 1/100 (0.01).
1/100 = 0.01The third place value to the right of the decimal point, representing 1/1000 (0.001).
1/1000 = 0.001The number being divided in a division problem.
The number by which the dividend is being divided.
The result obtained after performing division.
A decimal that contains a finite number of digits after the decimal point because the division reaches a remainder of zero.
A decimal whose digits continue infinitely with a repeating sequence of one or more digits because the remainders repeat in a cycle.
0.\overline{a_1 a_2 ... a_k}An integer of n digits with the property that when multiplied by 1, 2, 3, ..., n, the product contains the exact same digits in a cyclic permutation.
Multiplying both dividend and divisor by the same power of 10 to turn a decimal divisor into an integer before dividing.
(A × 10^k) ÷ (B × 10^k)The actual astronomical time taken by Earth to complete one revolution around the Sun (approximately 365.2422 days).
365.2422 daysA year containing 366 days instead of 365, with an extra day (February 29) added to keep the calendar aligned with Earth's revolution.
A year is a leap year if divisible by 4, except century years (divisible by 100), which are only leap years if divisible by 400.
A logic puzzle where numbers from 1 to N must be arranged in a continuous grid path connecting consecutive integers horizontally, vertically, or diagonally.
Frequently Asked Questions (FAQ)
What is a decimal?
A decimal is a way of writing numbers using place values based on powers of 10, extending beyond whole numbers into fractional parts: tenths (1/10), hundredths (1/100), thousandths (1/1000), separated from the whole-number part by a decimal point.
How do you multiply two decimals?
First, multiply the numbers as if there were no decimal points (as whole numbers). Then, count the total number of decimal digits across both factors. Finally, place the decimal point in the product counting that many digits from the right.
How do you divide a decimal by 10, 100, or 1000?
Dividing by 10 shifts the decimal point 1 place to the left; dividing by 100 shifts it 2 places to the left; and dividing by 1000 shifts it 3 places to the left. Add leading zeroes if needed (e.g., 12 ÷ 1000 = 0.012).
How do you divide when the divisor is a decimal?
Multiply both the dividend and the divisor by the same power of 10 (10, 100, 1000, etc.) so that the divisor becomes a whole number. Then perform standard division. For example, 4.68 ÷ 1.3 becomes 46.8 ÷ 13 = 3.6.
Why can we multiply both dividend and divisor by 10 without changing the answer?
A division A ÷ B is equivalent to the fraction A/B. Multiplying numerator and denominator by the same non-zero number (like 10/10) produces an equivalent fraction with the identical mathematical value.
Does multiplying always make a number larger?
No! Multiplying by a positive number between 0 and 1 makes the number smaller (e.g., 8 × 0.5 = 4). Multiplying by a number greater than 1 makes it larger. If both factors are between 0 and 1, the product is smaller than both.
Does division always make a number smaller?
No! Dividing by a positive number strictly between 0 and 1 makes the quotient larger than the dividend (e.g., 12 ÷ 0.4 = 30). This is because you are finding how many small fractional pieces fit inside the dividend.
What is a terminating decimal?
A terminating decimal is a decimal that comes to an end with a finite number of digits, because the remainder in long division eventually becomes zero (e.g., 1/4 = 0.25, 29/4 = 7.25).
What is a repeating (non-terminating) decimal?
A repeating decimal is a decimal whose digits continue infinitely in a repeating pattern because the division remainders begin to repeat in a cycle (e.g., 10 ÷ 3 = 3.333..., 1 ÷ 7 = 0.142857142857...).
Why does 1 ÷ 7 repeat?
When dividing by 7, the only possible non-zero remainders are 1, 2, 3, 4, 5, and 6. Because there are only 6 possible remainders, one of them must repeat within at most 6 division steps. Once a remainder repeats, the entire calculation repeats in a cycle!
What is the magic number 142857?
142857 is the repeating 6-digit block from 1/7. When multiplied by 1, 2, 3, 4, 5, and 6, the resulting products contain the exact same six digits in the same cyclic order (142857, 285714, 428571, 571428, 714285, 857142). When multiplied by 7, it gives 999,999!
What was Emil Artin's conjecture in 1927?
Austrian mathematician Emil Artin conjectured that there are infinitely many primes p whose fraction 1/p generates a full-period cyclic number like 142857. This remains an unsolved problem in modern number theory.
Why do we need leap years?
The Earth takes 365.2422 days to complete one revolution around the Sun, not 365 days. Without adjustments, our calendar year would drift by about 0.2422 days every year (over 24 days every century), causing seasons to drift across calendar months.
Why do we add a leap day every 4 years?
0.2422 days is close to 1/4 of a day (0.25). Adding 1 extra day every 4 years compensates for most of the drift (4 × 0.2422 = 0.9688 days, close to 1 day).
Why is 1900 not a leap year?
Adding a day every 4 years slightly overcompensates (0.25 vs 0.2422 days per year), building an extra 0.78 days every 100 years. To fix this, century years (divisible by 100) are NOT leap years unless they are also divisible by 400. 1900 is not divisible by 400, so it had only 365 days.
Why was 2000 a leap year?
Skipping every century year slightly undercorrects (by 2.2 days every 1000 years). The 400-year rule restores the leap day on century years divisible by 400. Since 2000 ÷ 400 = 5 with no remainder, 2000 was a leap year.
What is the complete Gregorian leap year rule?
A year is a leap year if it is divisible by 4, UNLESS it is divisible by 100, in which case it is only a leap year if it is ALSO divisible by 400.
Which fractions produce terminating decimals?
In simplest form, a fraction produces a terminating decimal if and only if the prime factorisation of its denominator contains ONLY 2s and/or 5s (the prime factors of 10).
What is a Hidato puzzle?
Hidato is a logic number puzzle played on a grid where you must fill consecutive numbers from 1 to N along an unbroken path, moving from each number to the next horizontally, vertically, or diagonally.
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