Another Peek Beyond the Point · Lesson 13 of 13
Chapter Summary and Practice
“Connect the chapter’s decimal ideas and apply them in mixed calculations, measurement, reasoning, and puzzles.”
• Connect decimal place value with fraction forms and powers of ten. • Choose and check multiplication or division in mixed situations. • Distinguish exact terminating and repeating answers from approximations. • Read decimal number lines and compare prices fairly using equal quantities. • Apply the calendar rule and solve mathematical pattern challenges.
Connect the ideas across the chapter
The central idea is place value. Decimal multiplication counts smaller units, while decimal division shares and regroups them. Fraction forms explain why the procedures work. An estimate tells you whether the size of an answer makes sense, and an inverse calculation checks whether the answer restores the original quantity. Use these connections to choose a method rather than treating every calculation as an isolated rule.
| Idea | Meaning or reliable method | Representative relationship |
|---|---|---|
| Place value | Each place to the right is one-tenth of the previous place | 0.254 = 2/10 + 5/100 + 4/1000 |
| Equivalent decimals | Trailing decimal zeroes preserve value | 9.5 = 9.500; 0.05 ≠ 0.5 |
| Multiply decimals | Multiply whole-number digit values, then divide by the combined powers of ten | 5.8 × 1.24 = 7192/1000 = 7.192 |
| Product size | For positive values, compare the multiplier with 1 | 8 × 0.5 < 8; 8 × 1 = 8; 8 × 2 > 8 |
| Powers of ten | Multiplication enlarges each digit value; division reduces it | 0.306 × 1000 = 306; 18.7 ÷ 100 = 0.187 |
| Equivalent fractions | Make a denominator 10, 100, 1000, or another power of ten when possible | 29/4 = 725/100 = 7.25 |
| Whole-number division | Regroup each remainder into the next smaller place | 1325 ÷ 4 = 331.25 |
| Decimal dividend | Share the original decimal places and continue if necessary | 0.06 ÷ 5 = 0.012 |
| Decimal divisor | Scale BOTH numbers equally until the divisor is whole | 4.68 ÷ 1.3 = 46.8 ÷ 13 = 3.6 |
| Quotient size | For a positive dividend, compare the positive divisor with 1 | 8 ÷ 0.5 > 8; 8 ÷ 1 = 8; 8 ÷ 2 < 8 |
| Exact decimal forms | Zero remainder ends division; repeated remainder makes digits repeat | 1/8 = 0.125; 1/7 = 0.142857… |
| Terminating patterns | Pair factors of 2 and 5 to make a power of ten | 1/16 = 625/10000 = 0.0625 |
| Measurements and rates | Keep units consistent and identify the quantity per unit | 125.5 mL = 0.1255 L; speed = distance ÷ time |
| Calendar investigation | Check accumulated differences and apply the ordered leap-year tests | Leap if divisible by 400, or by 4 but not by 100 |
Before starting a mixed problem, name the unknown quantity. Ask whether you are combining equal amounts, finding a share, counting equal groups, or calculating an amount per unit. After calculating, test the answer against an estimate and attach the correct unit. A numerical answer without its meaning can hide an operation mistake.
Worked mixed problems
The following problems connect several ideas at once. Price comparison requires an equal basis; shelf capacity requires whole objects rather than just a numerical quotient; mixed-number division can be checked using exact fractions. In each case, interpretation matters alongside arithmetic.
Problem
A 210 g packet of peanut chikki costs ₹70.50 and a 110 g packet of potato chips costs ₹33.25. Which has the lower price per 100 g?
- 1.The packet masses differ, so compare costs for the same mass rather than just the two total prices.
- 2.Chikki per 100 g costs (70.50 ÷ 210) × 100 = 7050/210 = 235/7 rupees, about ₹33.57.
- 3.Chips per 100 g cost (33.25 ÷ 110) × 100 = 3325/110 = 665/22 rupees, about ₹30.23.
- 4.The chips have the lower price per 100 g. These monetary decimals are rounded comparisons; the fractions give exact unit costs.
- 5.An exact check avoids rounding: 70.50 × 110 = 7755, whereas 33.25 × 210 = 6982.50. For positive masses, the smaller second cross-product confirms the lower chips unit price.
Problem
A shelf is 160 cm long. Each book is 2.5 cm thick. Can 80 books fit, and how much shelf space remains after the maximum possible number is placed?
- 1.The number that fits is 160 ÷ 2.5 = 1600 ÷ 25 = 64.
- 2.Check: 64 × 2.5 = 160 cm, so 64 whole books fit exactly with 0 cm left.
- 3.80 books need 80 × 2.5 = 200 cm, which is 40 cm longer than the shelf.
- 4.Thus 80 cannot fit; 16 of them must remain elsewhere. If a different quotient had a fractional part, only the whole number of complete books could be placed.
Problem
Find 6¼ ÷ 2½ and 60¼ ÷ 3½.
- 1.A mixed number combines a whole number and a fraction: 6¼ = 6 + 1/4 = 6.25; 2½ = 2.5.
- 2.6.25 ÷ 2.5 = 62.5 ÷ 25 = 2.5. Check: 2.5 × 2.5 = 6.25.
- 3.For the second division, 60¼ = 60.25 and 3½ = 3.5. Scale both: 60.25 ÷ 3.5 = 602.5 ÷ 35.
- 4.The exact fraction is (241/4) ÷ (7/2) = (241 × 2)/(4 × 7) = 241/14.
- 5.Its decimal is 17.214285714285…; the repeating digits begin after the first decimal digit. The finite value 17.214285 is only an approximation, not the exact quotient.
The two mixed-number divisions above are associated in the chapter with Sridharacharya’s Patiganita. Decimal forms and fraction forms let us approach the same old arithmetic questions in more than one way.
Problem
Shyamala buys 3 kg of bananas at ₹30 per kg. There are 35 bananas and she sells each for ₹5. Find her profit.
- 1.Her purchase cost is 3 × 30 = ₹90.
- 2.Her sales revenue is 35 × 5 = ₹175.
- 3.Profit is revenue minus purchase cost: 175 − 90 = ₹85.
- 4.Notice the different counting units: kilograms determine the purchase cost, while individual bananas determine the sales revenue. Do not multiply 3 kg by the selling price per banana.
Read decimal number lines and scaling tables
A number line’s equally spaced intervals represent equal differences. Count intervals, not just the interior tick marks. Divide the total difference between the labelled endpoints by the number of intervals to find the step. This works whether the endpoint difference is one tenth, two hundredths, or another decimal amount.
The division table below uses a across the top and b down the side. Each interior cell is a ÷ b. Start from 1517 ÷ 37 = 41 and track what changes. Reducing a alone to one-tenth reduces the quotient to one-tenth. Reducing b alone to one-tenth makes the quotient ten times as large. Reducing both together leaves the quotient unchanged.
| b ↓ / a → | 1517 | 151.7 | 15.17 | 1.517 | 15170 |
|---|---|---|---|---|---|
| 37 | 41 | ? | ? | ? | ? |
| 3.7 | ? | ? | 4.1 | ? | ? |
| 0.37 | ? | ? | ? | ? | ? |
| 0.037 | ? | 4100 | ? | ? | ? |
| 370 | ? | ? | ? | ? | ? |
Try a connected-number puzzle
A Hidato puzzle uses the consecutive whole numbers from 1 to the number of cells. Place each number once so that each consecutive pair occupies neighbouring cells. Neighbours may share a side or meet at a corner, so diagonal moves are allowed. The printed clues stay fixed. Although this is a whole-number puzzle, it develops the careful checking and pattern reasoning used throughout the chapter.
The grid below has 25 cells, so fill it with 1 through 25. Start with the short stretch from 1 to 4 and look for routes consistent with 8 and 9. Larger puzzles can use an irregular connected shape or holes; missing cells are never part of the path. A completed solution must pass all three tests: every required number appears once, every clue is preserved, and every consecutive pair is adjacent. Do not accept a sample path containing a duplicated or missing number.
For a further challenge, use the irregular 56-cell grid below. Only the outlined cells exist: the gaps around the narrow top and bottom extensions are outside the puzzle. The same side-or-corner adjacency rule applies, so a number cannot jump across a gap. The final number is 56 because there are exactly 56 available cells.
Between two fixed clues, count how many steps are available. A step can move at most one row and one column, so clues far apart may force a route. Try one short stretch at a time, write tentative entries in pencil, and check both ends before committing. Backtrack when a choice would strand a cell or prevent the next fixed clue from being reached.
Keep three different decimal rules separate: multiplication uses combined decimal-place counts; division with a decimal divisor scales both numbers equally; continuing long division uses remainders and place value. None of these justifies deleting all decimal points and guessing where the final point goes.
Quiz
Which calculation correctly finds 0.432 × 0.23?
Which division equals 2.46 ÷ 0.15?
What is the upper arrow position in the number-line diagram?
Why does the cheaper packet price alone give an unfair unit-price comparison?
What does a repeated non-zero remainder tell you when zeroes continue to be brought down?
Which statement about positive numbers is correct?
How many books 2.5 cm thick fit exactly on a 160 cm shelf?
Which century year is leap under the completed rule?
Which Hidato move is permitted between consecutive entries?
Practice Problems
- Write 23.047 in expanded form and as a fraction with denominator 1000. Explain the placeholder zero.
- Multiply 27.34 × 6, 4.23 × 3.7, and 0.432 × 0.23. Estimate before calculating.
- For three shirts needing 1.65 m of cloth each, find the total cloth. For a rectangle 5.7 cm by 13.3 cm, find its area.
- Calculate the cost of 2.250 kg of oranges at ₹56.50 per kg. Explain why 2.250 and 2.25 give the same result.
- Use 18 × 12 = 216 to find 1.8 × 1.2, 0.18 × 12, and 0.018 × 0.12.
- Calculate 18.7 ÷ 10, 18.7 ÷ 1000, 0.0058 ÷ 100, and 0.306 × 1000.
- Use long division to calculate 1325 ÷ 4 and 237 ÷ 8, naming the unit of each remainder.
- Calculate 9.5 ÷ 4, 0.06 ÷ 5, and 0.045 ÷ 3. Check each by multiplication.
- Calculate 4.68 ÷ 0.13 and 2.46 ÷ 1.5. Explain the scale factor used in each.
- Divide 13.5 kg of flour equally among 15 students; then divide 3 L of juice among eight friends. Express the juice share in litres and millilitres.
- Find the fuel efficiency for 234.45 km travelled with 12.6 L of fuel. Give an exact fraction or quotient and a clearly labelled approximation to four decimal places.
- Write 1/7 and 10/3 using repeating notation. Explain what would be wrong with claiming their first six decimal digits are exact finite answers.
- Find 1/16, 1/32, and 1/625 using equivalent decimal fractions. Describe the role of 2 × 5 = 10.
- Classify 1900, 2000, 2024, and 2100. Find the model difference for calendar years 1 through 10000.
- Read both arrow positions in the number-line diagram. Show the total interval length, step size, and number of steps.
- Convert 5.5 km to m, 35 cm to m, 14.5 cm to mm, 68 g to kg, 9.02 m to mm, and 125.5 mL to L.
- Complete every blank in the a ÷ b table. Explain one row pattern and one column pattern.
- Find 6¼ ÷ 2½ and 60¼ ÷ 3½ by both decimal and fraction methods. Distinguish the terminating answer from the repeating answer.
- Give at least two different pairs of positive factors with product 2.4, and two with product 14.5.
- Using 756 ÷ 36 = 21, find 75.6 ÷ 3.6, 7.56 ÷ 0.36, 756 ÷ 0.36, 75.6 ÷ 360, and 7560 ÷ 3.6.
- Use the digits 2, 4, 5, 8, and 0 exactly once in □□.□ × □.□. Find the greatest and least products, a product greater than 150, a product nearest to 100, and a product nearest to 5. State whether you allow 0 in the first box, and investigate both conventions.
- Arrange these six values in increasing order without calculating all products and quotients fully: 245.05 × 0.942368; 245.05 × 7.9682; 245.05 ÷ 7.9682; 245.05 ÷ 0.942368; 245.05; 7.9682. Justify any comparison needing more than the above-or-below-1 rule.
- Solve the Hidato grid and check all consecutive pairs, including diagonal moves.
- Solve the irregular 56-cell Hidato challenge. Confirm that every outlined cell is used and that no step jumps across an absent cell.
- A shop buys 50 notebooks at ₹23.60 each and sells them at ₹30 each. Find total cost, total revenue, and profit. Explain how both multiplication and subtraction are used.
Rows from top to bottom: 25, 23, 1, 2, 3; 24, 20, 22, 5, 4; 19, 14, 21, 8, 6; 18, 13, 15, 7, 9; 17, 16, 12, 11, 10. Check every pair from 1–2 through 24–25, rather than checking only the clue cells.
Key Takeaways
• Place value and equivalent fractions explain the chapter’s decimal procedures. • Predict answer size by comparing a positive multiplier or divisor with 1. • Regroup long-division remainders carefully and preserve zero placeholders. • Scale dividend and divisor equally when making a decimal divisor whole. • Use repeating notation for exact non-terminating decimals and label finite approximations. • Compare quantities on an equal unit basis and keep units throughout a calculation. • Check formulas, number-line steps, calendar counts, and puzzle paths rather than relying on appearances.
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Look Before You Leap!
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