Question decimal-expansion
Write 27.53 as a sum of place-value parts.
Steps only
2 tens=20; 7 ones=7; 5 tenths=0.5; 3 hundredths=0.03.
Add the parts.
Answer
27.53=20+7+0.5+0.03.
CLASS 7 · GANITA PRAKASH PART II · CHAPTER 4
Another Peek Beyond the Point
Extend place value to multiply and divide decimals, interpret quotients, and model calendar adjustments with decimals.
68 worked answersAll 4 exercise sets
What you will learn
- Represent decimals as sums of place-value units.
- Multiply and divide decimals accurately using place value.
- Estimate whether decimal products and quotients are larger or smaller.
- Use decimal arithmetic in measurement, pricing and calendar problems.
68 worked answers · short steps first, explanations when you need them
4.1 Decimal place value
Textbook pages 68–69
Question spices-in-kilograms
Convert 50 g cinnamon, 100 g cumin, 25 g cardamom and 250 g pepper to kilograms as fractions and decimals.
Steps only
1 kg=1,000 g, so divide each mass by 1,000.
50/1,000=1/20; 100/1,000=1/10; 25/1,000=1/40; 250/1,000=1/4.
Answer
50 g=1/20 kg=0.05 kg; 100 g=1/10 kg=0.1 kg; 25 g=1/40 kg=0.025 kg; 250 g=1/4 kg=0.25 kg.
Question decimal-fraction-expansions
Expand 254/1,000, 847/10,000, 173/100 and 23/1,000, then write their decimals.
Steps only
254/1,000=2/10+5/100+4/1,000.
847/10,000=8/100+4/1,000+7/10,000.
173/100=1+7/10+3/100.
23/1,000=2/100+3/1,000.
Answer
0.254, 0.0847, 1.73 and 0.023, respectively.
Question divide-powers-of-ten
Evaluate 123÷10, 24÷100, 678÷1,000, 12÷1,000 and 12,345÷1,000.
Steps only
Move the decimal point left as many places as there are zeroes in the divisor.
Add leading zeroes when there are not enough whole-number digits.
Answer
12.3, 0.24, 0.678, 0.012 and 12.345.
4.2 Decimal multiplication
Textbook pages 69–73
Question five-pens
One pen costs ₹9.50. Find the cost of 5 pens.
Steps only
9.50×5=47.50.
Check by adding ₹9.50 five times.
Answer
₹47.50.
Question car-distance
A car travels 12.5 km per litre. How far does it travel on 7.5 litres?
Steps only
12.5×7.5=(125/10)(75/10).
125×75=9,375, then divide by 100.
Answer
93.75 km.
Question weekly-walk
Ajay’s school is 827 m away. He walks there and back 6 days a week. Find the weekly distance in kilometres.
Steps only
One-way distance=0.827 km.
Daily round trip=2×0.827=1.654 km.
Weekly distance=6×1.654.
Answer
9.924 km.
Question rectangle-area-decimals
Find the area of a rectangle 5.7 cm by 13.3 cm.
Steps only
Area=5.7×13.3.
57×133=7,581; there are two decimal places altogether.
Answer
75.81 cm².
Question decimal-product-whole
Can a product of two decimals, or a decimal and a whole number, be a whole number? Give examples.
Steps only
2.5×4=10 is a decimal times a whole number giving a whole number.
0.5×0.2=0.1 shows two decimals need not give a whole number.
Answer
Yes, they can, but they do not always. For example 1.25×0.8=1.
Question decimal-product-rule-example
Given 596×248=147,808, find 5.96×24.8.
Steps only
Ignore decimal points first: product digits are 147,808.
There are 2+1=3 decimal places in the factors.
Place the decimal three digits from the right.
Answer
147.808.
Question decimal-product-size
Compare the product with its factors when (i) both factors exceed 1, (ii) both lie between 0 and 1, or (iii) one is below 1 and one above 1.
Steps only
Multiplying by a number above 1 increases a positive number.
Multiplying by a positive number below 1 decreases a positive number.
If one factor is above 1 and one below 1, compare the product separately with each factor.
Answer
- (i) Product is greater than both;
- (ii) less than both;
- (iii) greater than the factor below 1 and less than the factor above 1.
Question tenths-hundredths-products
Find (a) 6×4 tenths, (b) 7×0.3 and (c) 9×5 hundredths.
Steps only
4 tenths=0.4; 6×0.4=2.4.
7×0.3=21 tenths=2.1.
5 hundredths=0.05; 9×0.05=45 hundredths=0.45.
Answer
(a) 2.4, (b) 2.1, (c) 0.45.
Question products-exercise
Find (a) 27.34×6, (b) 4.23×3.7 and (c) 0.432×0.23.
Steps only
(a) 2,734×6=16,404; place 2 decimal digits.
(b) 423×37=15,651; place 3 decimal digits.
(c) 432×23=9,936; place 5 decimal digits.
Answer
(a) 164.04, (b) 15.651, (c) 0.09936.
Question cloth-three-shirts
One shirt needs 1.65 m of cloth. How much is needed for 3 shirts?
Steps only
1.65×3=4.95.
Answer
4.95 m.
Question notebooks-erasers-cost
Meenu buys 4 notebooks at ₹15.50 each and 3 erasers at ₹2.75 each. Find the total.
Steps only
Notebooks: 4×15.50=₹62.00.
Erasers: 3×2.75=₹8.25.
Add 62+8.25.
Answer
₹70.25.
Question coin-stack-thickness
A rupee coin is 1.45 mm thick. Find the height of 36 coins in centimetres.
Steps only
36×1.45=52.2 mm.
10 mm=1 cm, so 52.2 mm=5.22 cm.
Answer
5.22 cm.
Question orange-price
Oranges cost ₹56.50 per kilogram. Find the cost of 2.250 kg. Does writing 56.50 as 56.5 and 2.250 as 2.25 change the answer?
Steps only
56.5×2.25=56.5×(2+0.25)=113+14.125.
Trailing zeroes after the last non-zero decimal digit do not change a value.
Answer
₹127.125 (₹127.13 to the nearest paisa). Writing 56.50 as 56.5 and 2.250 as 2.25 gives the same product.
Question wholesale-notebook-profit
A shopkeeper buys notebooks at ₹23.60 each and sells them at ₹30 each. What is the profit on 50 notebooks?
Steps only
Profit per notebook=30−23.60=₹6.40.
50×6.40=₹320.
Answer
₹320.
Question scale-products-18-12
Given 18×12=216, find (a) 18×1.2, (b) 18×0.12, (c) 1.8×1.2, (d) 0.18×0.12, (e) 0.018×0.012 and (f) 1.8×12. Which are below 1?
Steps only
Use 18×12=216 and account for decimal-place shifts.
(a) one decimal place total: 21.6; (b) two: 2.16; (c) two: 2.16.
(d) four: 0.0216; (e) six: 0.000216; (f) one: 21.6.
Answer
21.6, 2.16, 2.16, 0.0216, 0.000216, 21.6. Only (d) and (e) are below 1.
Question predict-product-below-one
Which products are below 1: (a) 7×0.6, (b) 0.7×0.6, (c) 0.7×6, (d) 0.07×0.06?
Steps only
(a) 7×a number below 1 can still exceed 1; 7×0.6=4.2.
(b) two positive numbers below 1 give a product below either: below 1.
(c) 0.7×6=4.2, above 1.
(d) 0.07×0.06 is much smaller than 1.
Answer
Only (b) and (d) are below 1.
Question multiply-by-powers-table
Complete products by 10, 100 and 1,000 for 5.7, 23.02, 0.92, 0.306 and 24.67.
Steps only
Multiplying by 10, 100 and 1,000 moves the decimal point right by 1, 2 and 3 places.
Answer
5.7→57; 570; 5,700. 23.02→230.2; 2,302; 23,020. 0.92→9.2; 92; 920. 0.306→3.06; 30.6; 306. 24.67→246.7; 2,467; 24,670.
4.3 Decimal division
Textbook pages 74–87
Question ribbon-divided-ten-hundred
A 3.9 m ribbon is cut into 10 equal pieces, then imagine 100 equal pieces. Find each piece length.
Steps only
3.9÷10=0.39 m.
3.9÷100=0.039 m.
0.039 m=3.9 cm=39 mm.
Answer
For 10 pieces: 0.39 m each. For 100 pieces: 0.039 m each (3.9 cm or 39 mm).
Question divide-by-powers-table
Complete ÷10, ÷100, ÷1,000 and ÷10,000 for 18.7, 21.1, 0.13, 2.146 and 0.0058.
Steps only
Move the decimal point left by the number of zeroes.
Write leading zeroes when needed.
Answer
18.7→1.87, 0.187, 0.0187, 0.00187; 21.1→2.11, 0.211, 0.0211, 0.00211; 0.13→0.013, 0.0013, 0.00013, 0.000013; 2.146→0.2146, 0.02146, 0.002146, 0.0002146; 0.0058→0.00058, 0.000058, 0.0000058, 0.00000058.
Question ribbon-shared-two-four
Share 29 m of ribbon equally between 2 friends, then among 4 friends.
Steps only
Question long-division-1324
Find 1,324÷4 using place-value division.
Steps only
1 thousand regrouped as 10 hundreds gives 13 hundreds; each share gets 3 hundreds, remainder 1 hundred.
Regroup the remaining hundred as 10 tens: 12 tens gives 3 tens each.
4 ones gives 1 one each.
Answer
331.
Question long-division-1325
Find 1,325÷4 by extending place-value division.
Steps only
1,324÷4=331, with 1 left over.
Regroup 1 one as 10 tenths: each gets 2 tenths, remainder 2 tenths.
Regroup 2 tenths as 20 hundredths: each gets 5 hundredths.
Answer
331.25.
Question long-division-237-8
Divide 237 by 8 to thousandths.
Steps only
8 goes into 237 twenty-nine times, remainder 5.
Regroup 5 ones as 50 tenths: 6 tenths, remainder 2.
Regroup 2 tenths as 20 hundredths: 2 hundredths, remainder 4.
Regroup 4 hundredths as 40 thousandths: 5 thousandths.
Answer
29.625.
Question sugar-four-bags
A shopkeeper has 9.5 kg of sugar and packs it equally into 4 bags. How much per bag?
Steps only
9.5÷4=95/10÷4=95/40=19/8.
19/8=2.375.
Answer
2.375 kg per bag.
Question six-hundredths-divided-five
Find 0.06÷5.
Steps only
6 hundredths divided by 5 gives 1 hundredth each, remainder 1 hundredth.
Regroup that as 10 thousandths; 10÷5=2 thousandths.
Answer
0.012.
Question division-fractions-figure
Convert 18/5, 415/4, 1217/2 and 4827/8 to decimal form, then verify by multiplication.
Steps only
18/5=36/10=3.6. For 415/4, make denominator 100: 415×25/100=10,375/100.
1217/2=608.5. For 4827/8, multiply top and bottom by 125 to make denominator 1,000: 603,375/1,000.
Check by multiplying each quotient by its divisor to recover the numerator.
Answer
18/5=3.6; 415/4=103.75; 1217/2=608.5; 4827/8=603.375.
Question quotient-exercise-fractions
Write 2/5, 13/4, 4/50 and 5/8 in decimal form.
Steps only
2/5=4/10=0.4; 13/4=325/100=3.25.
4/50=8/100=0.08; 5/8=625/1,000=0.625.
In each case, multiply the top and bottom by the same number so the denominator is a power of 10.
Answer
The decimals are 0.4, 3.25, 0.08 and 0.625, respectively.
Question decimal-division-exercise
Find (a) 24.86÷1.2 and (b) 5.728÷1.52.
Steps only
(a) Multiply both numbers by 10: 248.6÷12=1,243/60=20.7166… .
(b) Multiply both numbers by 100: 572.8÷152=358/95=3.768421… .
Answer
(a) 20.7166…; (b) 3.768421… (recurring).
Question select-quotients
Choose the correct values for 1,526÷4 and 3,567÷8.
Steps only
1,526÷4=381.5.
3,567÷8=445 remainder 7; 7/8=0.875.
Answer
1,526÷4=381.5 (option iii); 3,567÷8=445.875 (option iii).
Question 132-scale-divisions
Find 132÷4, 13.2÷4, 1.32÷4 and 0.132÷4.
Steps only
132÷4=33.
Each tenfold reduction in the dividend gives a tenfold reduction in the quotient.
Answer
33, 3.3, 0.33 and 0.033.
Question 126-scale-divisions
Find 126÷8, 12.6÷8, 1.26÷8, 0.126÷8 and 0.0126÷8.
Steps only
126÷8=15.75.
Move the decimal left one place in each successive dividend.
Answer
15.75, 1.575, 0.1575, 0.01575 and 0.001575.
Question decimal-divisor-speed
Ravi travels 126 km in 2.5 hours. Find his average speed.
Steps only
126÷2.5=(126×10)÷(2.5×10)=1,260÷25.
1,260÷25=50.4.
Answer
50.4 km/h.
Question decimal-divisor-468
Find 4.68÷1.3 and 4.68÷0.13.
Steps only
4.68÷1.3=46.8÷13=3.6.
4.68÷0.13=468÷13=36.
Answer
3.6 and 36.
Question nonterminating-tenths-thirds
Show why 10÷3 has a repeating decimal.
Steps only
10÷3=3 remainder 1.
At each decimal place, regroup the remainder 1 into 10 of the next unit; 10÷3 again gives 3 remainder 1.
The same remainder repeats forever.
Answer
10÷3=3.333… and does not terminate.
Question fractions-repeating
Find 10÷9, 100÷11 and 1÷7. Which decimal digits repeat?
Steps only
10÷9=1 remainder 1, so the digit 1 repeats.
100÷11=9 remainder 1; then 10÷11 gives 0 remainder 10, then 100÷11 gives 9 again.
Long division of 1÷7 produces remainder cycle 1,3,2,6,4,5.
Answer
10÷9=1.111…; 100÷11=9.0909…; 1÷7=0.142857142857… .
Question cyclic-number-142857
What happens when 142857 is multiplied by 1 through 7?
Steps only
The products for 1–6 are cyclic rotations: 142857, 285714, 428571, 571428, 714285, 857142.
142857×7=999999.
Answer
Multiplication by 1–6 cycles the digits; multiplication by 7 gives 999,999. The repeating block comes from 1/7.
Question quotient-divisor-size
Can dividing by a decimal make the quotient bigger than the dividend? Give the rule for positive values.
Steps only
128÷0.4=320, which is greater than 128.
Dividing by a number below 1 is equivalent to multiplying by its reciprocal, which is above 1.
Dividing by a number above 1 makes a positive quotient smaller than the dividend.
Answer
Yes. For positive numbers, divisor<1 makes quotient>dividend; divisor>1 makes quotient<dividend; divisor=1 leaves it unchanged.
Question known-156-division-questions
Find (a) 15.6×1.2, (b) 187.2÷1.2, (c) 18.72÷15.6 and (d) 0.156×0.12 using 156×12=1,872.
Steps only
(a) The factors are each 10 times smaller than 156 and 12, so the product is 18.72.
(b) 187.2÷1.2=1,872÷12=156. (c) 18.72÷15.6=187.2÷156=1.2.
(d) 0.156×0.12 has five decimal places; use 1,872 and place the decimal to get 0.01872.
Answer
(a) 18.72; (b) 156; (c) 1.2; (d) 0.01872.
Question fill-division-blanks
Complete: (a) 25÷□=0.025; (b) 25÷□=250; (c) 25÷□=2.5; (d) 25÷10=25×□; (e) 25÷0.10=25×□; (f) 25÷0.01=25×□.
Steps only
For (a)–(c), divisor=25÷quotient: 1,000; 0.1; and 10.
Dividing by a number is multiplying by its reciprocal.
So 1/10=0.1, 1/0.10=10 and 1/0.01=100.
Answer
(a) 1,000; (b) 0.1; (c) 10; (d) 0.1; (e) 10; (f) 100.
Question scaled-246-quotients
Find 2.46÷1.5, 2.46÷0.15 and 2.46÷0.015. Is 24.6÷1.5 equal to 2.46÷0.15?
Steps only
Make each divisor a whole number: 2.46÷1.5=24.6÷15=1.64.
2.46÷0.15=246÷15=16.4; 2.46÷0.015=2,460÷15=164.
24.6÷1.5=246÷15=16.4. Compare with the second quotient.
Answer
The quotients are 1.64, 16.4 and 164. Yes, 24.6÷1.5 and 2.46÷0.15 are both 16.4.
Question wooden-block-sharing
A 4 m block is cut into 5 equal pieces. Find each piece’s length.
Steps only
Divide total length by number of pieces: 4÷5=4/5 m.
4/5=8/10=0.8 m.
Answer
Each piece is 0.8 m long (80 cm).
Question regular-polygon-side
A regular 12-sided polygon has perimeter 208.8 cm. Find the length of one side.
Steps only
All 12 sides are equal.
Side length=208.8÷12=17.4.
Answer
Each side is 17.4 cm long.
Question watermelon-juice-sharing
Share 3 L of watermelon juice equally among 8 friends. How many millilitres does each friend get?
Steps only
3 L=3,000 mL.
3,000÷8=375.
Answer
Each friend gets 375 mL.
Question car-distance-per-litre
A car covers 234.45 km using 12.6 L of petrol. How far does it travel per litre?
Steps only
Distance per litre=234.45÷12.6.
Multiply both numbers by 10: 2,344.5÷126. Or use 23,445÷1,260=521/28.
521/28=18.607142… .
Answer
About 18.61 km per litre (exactly 521/28 km/L).
Question flour-equal-share
Divide 13.5 kg of flour equally among 15 students. How much does each receive?
Steps only
Decimal applications and leap years
Textbook pages 87–95
Question peanut-vs-chips-unit-price
A 210 g peanut-chikki packet costs ₹70.50 and a 110 g chips packet costs ₹33.25. Which is cheaper per gram?
Steps only
Chikki: 70.50÷210≈₹0.3357 per gram.
Chips: 33.25÷110≈₹0.3023 per gram.
Answer
The potato chips are cheaper per gram.
Question arrow-decimal-number-line
Write the decimals shown by arrows between 3.1 and 3.2 and between 2.15 and 2.17.
Steps only
The first arrow is at the sixth hundredth after 3.1: 3.16.
The second arrow is at the midpoint between 2.15 and 2.17, so it is 2.16.
Answer
The arrow values are 3.16 and 2.16.
Question banana-profit
Shyamala buys 3 kg of bananas at ₹30/kg. There are 35 bananas and she sells each for ₹5. Find her profit if all are sold.
Steps only
Purchase cost=3×30=₹90.
Revenue=35×5=₹175.
Profit=175−90.
Answer
₹85 profit.
Question bookshelf-capacity
Each textbook is 2.5 cm thick. The shelf is 160 cm long. How many of 80 books fit, and how much space remains?
Steps only
160÷2.5=1,600÷25=64 books.
64×2.5=160 cm.
Answer
64 books fit; no shelf space remains. The remaining 16 books do not fit on this shelf.
Question metric-unit-conversions
Fill: 5.5 km to m; 35 cm to m; 14.5 cm to mm; 68 g to kg; 9.02 m to mm; 125.5 mL to L.
Steps only
Use 1 km=1,000 m, 1 m=100 cm=1,000 mm, 1 kg=1,000 g and 1 L=1,000 mL.
Move the decimal according to the unit scale.
Answer
5,500 m; 0.35 m; 145 mm; 0.068 kg; 9,020 mm; 0.1255 L.
Question sridharacharya-division
Evaluate 6 1/4÷2 1/2 and 60 1/4÷3 1/2 using decimals.
Steps only
6 1/4=6.25 and 2 1/2=2.5; 6.25÷2.5=2.5.
60 1/4=60.25 and 3 1/2=3.5; 60.25÷3.5=602.5÷35=17.214285… .
Answer
2.5 and 17.214285… (exactly 241/14).
Question make-products-2-4-14-5
Fill each pair of boxes in at least two ways: product 2.4 and product 14.5.
Steps only
For 2.4, 0.6×4=2.4 and 1.2×2=2.4.
For 14.5, 2.9×5=14.5 and 1.45×10=14.5.
Answer
Examples: 0.6×4 or 1.2×2; 2.9×5 or 1.45×10.
Question scale-quotients-from-756
Given 756÷36=21, find (a) 75.6÷3.6, (b) 7.56÷0.36, (c) 756÷0.36, (d) 75.6÷360, (e) 7,560÷3.6 and (f) 7.56÷0.36.
Steps only
If dividend and divisor are scaled equally, the quotient stays 21.
For (c) the divisor is 100 times smaller, so quotient is 100 times larger.
For (d) the dividend is 10 times smaller and divisor 10 times larger: quotient is 100 times smaller.
For (e), dividend is 10 times larger and divisor 10 times smaller: quotient is 100 times larger.
Answer
(a) 21, (b) 21, (c) 2,100, (d) 0.21, (e) 2,100, (f) 21.
Question division-table-missing-cells
Each table entry is a÷b. Fill the missing quotients for a=1,517; 151.7; 15.17; 1.517; 15,170 and b=37; 3.7; 0.37; 0.037; 370.
Steps only
Divide each column heading a by its row value b. Use 1,517÷37=41 as a starting fact.
Changing a by a factor of 10 changes the quotient by 10; changing b by a factor of 10 changes it by a factor of 1/10.
Across b=37, the quotients are 41, 4.1, 0.41, 0.041, 410. Continue the same scaling for each row.
Answer
Completed rows (in column order 1,517; 151.7; 15.17; 1.517; 15,170): b=37 → 41, 4.1, 0.41, 0.041, 410; b=3.7 → 410, 41, 4.1, 0.41, 4,100; b=0.37 → 4,100, 410, 41, 4.1, 41,000; b=0.037 → 41,000, 4,100, 410, 41, 410,000; b=370 → 4.1, 0.41, 0.041, 0.0041, 41.
Question decimal-digit-boxes-product
Fill □□.□×□.□ using 2, 4, 5, 8 and 0 once each. Find the maximum, minimum, one product above 150, and products nearest 100 and 5.
Steps only
Maximum: 82.0×5.4=442.8. Minimum: 45.8×0.2=9.16.
Above 150: 54.0×2.8=151.2. Near 100: 20.5×4.8=98.4, a distance of 1.6.
The smallest product is already 9.16, so it is also nearest to 5 (distance 4.16). Checking all 120 arrangements confirms the maximum, minimum and nearest examples.
Answer
Examples: maximum 82.0×5.4=442.8; minimum 45.8×0.2=9.16; above 150: 54.0×2.8=151.2; nearest 100: 20.5×4.8=98.4; nearest 5: 45.8×0.2=9.16.
Question sort-decimal-expressions
Sort: (a) 245.05×0.942368, (b) 245.05×7.9682, (c) 245.05÷7.9682, (d) 245.05÷0.942368, (e) 245.05, (f) 7.9682.
Steps only
Approximate: (f)=7.9682; (c)≈30.7535; (a)=230.9273.
(e)=245.05; (d)≈260.0364; (b)≈1,952.6074.
Answer
Increasing order: (f), (c), (a), (e), (d), (b).
Question leap-year-century
Why is a leap day added roughly every four years, but omitted in a century year? Which month receives the extra day?
Steps only
Earth’s orbit takes about 365.2422 days.
Using 365 days leaves about 0.2422 day each year; after four years this is about 0.9688 day.
Adding one day every four years corrects most of the drift, but slightly overcorrects over centuries.
Answer
The extra day is added to February, making 29 days. A leap day every four years corrects most of the fraction of a day omitted each ordinary year; century exceptions reduce the small overcorrection.
Question days-in-four-years
With a leap day every fourth year, how many calendar days are in four years? Compare this with the Earth’s four-orbit estimate at 365.2422 days per orbit.
Steps only
Calendar count=4×365+1=1,461 days.
Orbital estimate=4×365.2422=1,460.9688 days.
Difference=1,461−1,460.9688=0.0312 day.
Answer
The calendar has 1,461 days, which is 0.0312 day more than the 1,460.9688-day orbital estimate.
Question indian-calendar-research
How might you investigate how traditional calendars in India keep calendar dates aligned with seasonal or astronomical events?
Steps only
Choose one calendar system and use a library book, teacher-approved reference or reliable museum/educational source. Record the source and the calendar’s name.
Find out what observations or repeating cycles it uses, how it adjusts dates or months, and which seasonal or astronomical event it tracks.
Compare the calendar rule with one example date or event, then share the evidence and any limits in a short table.
Answer
This is a research activity, so the result depends on the calendar selected and the sources consulted. A good response names its source, explains the adjustment rule in the student’s own words and gives one example of how the rule keeps dates aligned.
Question days-in-100-years
Under the simple rule of one leap day every four years, how many calendar days are in 100 years? Compare with 100×365.2422.
Steps only
Calendar count=100×365+25=36,525 days.
Orbital time=100×365.2422=36,524.22 days.
Difference=0.78 day.
Answer
The simple rule gives 36,525 days, about 0.78 day more than 36,524.22 days.
Question days-in-100-years-skip-century
If the 100th year is not a leap year, how many days are in the 100-year period?
Steps only
There are 25 multiples of 4 from 1 to 100, but exclude year 100: 24 leap days.
Days=100×365+24.
Answer
36,524 days, which is 0.22 day less than 100×365.2422=36,524.22.
Question days-in-1000-years
How many calendar days are counted in 1,000 years under the rule: leap every 4 years, skip centuries unless divisible by 400?
Steps only
Leap years divisible by 4: 250. Century years: 10; of these, 2 are divisible by 400.
Ordinary years=750; leap years not century=240; century non-leap=8; 400-year leap=2.
Days=750×365+240×366+8×365+2×366.
Answer
365,242 days. The stated orbital estimate gives 365,242.2 days, so this calendar count is 0.2 day shorter.
Question days-in-10000-years
Using the rule divisible by 4, except centuries unless divisible by 400, find calendar days in 10,000 years and compare with 365.2422 days per year.
Steps only
Leap days=10,000/4−10,000/100+10,000/400=2,500−100+25=2,425.
Calendar days=10,000×365+2,425=3,652,425.
Orbital estimate=10,000×365.2422=3,652,422.
Answer
Calendar: 3,652,425 days; orbital estimate: 3,652,422 days. The calendar has 3 more days over 10,000 years, so one possible correction is to omit about three leap days in that span.
Question pattern-reciprocal-powers
Complete the next terms in (1/2), (1/2)×2, (1/2)×2×2, … and the corresponding sequence starting at 1/5. Explain the pattern.
Steps only
For 1/2: 0.5, 0.25, 0.125, 0.0625, next 0.03125.
For 1/5: 0.2, 0.04, 0.008, 0.0016, next 0.00032.
Answer
The next values are 0.03125 and 0.00032. Each step multiplies the preceding value by 1/2 or 1/5, respectively.