Another Peek Beyond the Point

Investigate how traditional calendars in India managed to consistently align the days in the calendar with astronomical events like the Earth going around the Sun or even the positions of the stars in the sky accurately.

Try solving the following Hidato puzzles.

Recall that a tenth is 0.1, a hundredth is 0.01, and so on. Find the following products in tenths, hundredths and so on:

(a) 6 × 4 tenths = 24 tenths (b) 7 × 0.3 (c) 9 × 5 hundredths

Find the products:

(a) $27.34 \times 6$

(b) $4.23 \times 3.7$

(c) $0.432 \times 0.23$

Thejus needs 1.65 m of cloth for a shirt. How many metres of cloth are needed for 3 shirts?

Meenu bought 4 notebooks and 3 erasers. The cost of each book was ₹15.50 and each eraser was ₹2.75. How much did she spend in all?

The thickness of a rupee coin is 1.45 mm. What is the total height of the cylinder formed by placing 36 rupee coins one over the other? Write the answer in centimeters.

The price of 1 kg of oranges is ₹56.50. What is the price of 2.250 kg of oranges? Can we write 56.50 as 56.5 and 2.250 as 2.25 and multiply? Will we get the same product? Why?

Dwarakanath purchases notebooks at a wholesale price of ₹23.6 per piece and sells each notebook at ₹30/-. How much profit does he make if he sells 50 books in a week?

Given that 18 × 12 = 216, find the products:

(a) 18 × 1.2 (b) 18 × 0.12 (c) 1.8 × 1.2 (d) 0.18 × 0.12 (e) 0.018 × 0.012 (f) 1.8 × 12

In which of the cases above is the product less than 1?

In which of the following multiplications is the product less than 1? Can you find the answer without actually doing the multiplications?

(a) $7 \times 0.6$

(b) $0.7 \times 0.6$

(c) $0.7 \times 6$

(d) $0.07 \times 0.06$

Multiplying the following numbers by 10, 100 and 1000 to complete the table.

Find the quotient by converting the denominator into 1, 10, 100 or 1000 and verify the solution by the long division method (division by place value).

(a) $\frac{18}{5}$ (b) $\frac{415}{4}$ (c) $\frac{1217}{2}$ (d) $\frac{4827}{8}$

Choose the correct answer:

(a) $\frac{1526}{4} =$

(i) 38.15

(ii) 380.15

(iii) 381.5

(iv) 381.05

(b) $\frac{3567}{8} =$

(i) 4458.75

(ii) 44.5875

(iii) 445.875

(iv) 4458.75

What is the quotient?

(a) $132 \div 4 =$

(b) $13.2 \div 4 =$

(c) $1.32 \div 4 =$

(d) $0.132 \div 4 =$

What is the quotient?

(a) $126 \div 8 =$

(b) $12.6 \div 8 =$

(c) $1.26 \div 8 =$

(d) $0.126 \div 8 =$

(e) $0.0126 \div 8 =$

Express the following fractions in decimal form:

(a) $\frac{2}{5}$

(b) $\frac{13}{4}$

(c) $\frac{4}{50}$

(d) $\frac{5}{8}$

Find the quotients:

(a) $24.86 \div 1.2$

(b) $5.728 \div 1.52$

Evaluate the following using the information $156 \times 12 = 1872$:

(a) $15.6 \times 1.2 = \underline{\quad\quad\quad\quad}$

(b) $187.2 \div 1.2 = \underline{\quad\quad\quad\quad}$

(c) $18.72 \div 15.6 = \underline{\quad\quad\quad\quad}$

(d) $0.156 \times 0.12 = \underline{\quad\quad\quad\quad}$

Evaluate the following:

(a) $25 \div \underline{\quad\quad} = 0.025$

(b) $25 \div \underline{\quad\quad} = 250$

(c) $25 \div \underline{\quad\quad} = 2.5$

(d) $25 \div 10 = 25 \times \underline{\quad\quad}$

(e) $25 \div 0.10 = 25 \times \underline{\quad\quad}$

(f) $25 \div 0.01 = 25 \times \underline{\quad\quad}$

Find the quotient:

(a) $2.46 \div 1.5 =$

(b) $2.46 \div 0.15 =$

(c) $2.46 \div 0.015 =$

Is the quotient obtained in $24.6 \div 1.5$ the same as the quotient obtained in $2.46 \div 0.15$?

A 4 m long wooden block has to be cut into 5 pieces of equal length. What is the length of each piece?

If the perimeter of a regular polygon with 12 sides is 208.8 cm, what is the length of its side?

3 litres of watermelon juice is shared among 8 friends equally. How much watermelon juice will each get? Express the quantity of juice in millilitres.

A car covers 234.45 km using 12.6 litres of petrol. What is the distance travelled per litre?

5 kg of flour (aata) was distributed equally among 15 students. How much flour did each student receive?

A 210 gram packet of peanut chikki costs ₹70.5, while a 110 gram packet of potato chips costs ₹33.25. Which is cheaper?

Write the decimal number at the arrow mark:

Shyamala bought 3 kg bananas at ₹30/- per kg. She counted 35 bananas in all. She sells each banana for ₹5/-. How much profit does she make selling all the bananas?

A teacher placed textbooks that are 2.5 cm thick on a bookshelf. The teacher wanted to place 80 textbooks on the shelf. The bookshelf is 160 cm long. How many books could be placed on the shelf? Was there any space left? If yes, how much?

Fill in the following blanks appropriately:

(a) $5.5\text{ km} = \underline{\hspace{1.5cm}}\text{ m}$

(b) $35\text{ cm} = \underline{\hspace{1.5cm}}\text{ m}$

(c) $14.5\text{ cm} = \underline{\hspace{1.5cm}}\text{ mm}$

(d) $68\text{ g} = \underline{\hspace{1.5cm}}\text{ kg}$

(e) $9.02\text{ m} = \underline{\hspace{1.5cm}}\text{ mm}$

(f) $125.5\text{ ml} = \underline{\hspace{1.5cm}}\text{ l}$

The following problem was set by Sridharacharya in his book, Patiganita. “$6 \frac{1}{4}$ is divided by $2 \frac{1}{2}$, and $60 \frac{1}{4}$ is divided by $3 \frac{1}{2}$. Tell the quotients separately.” Can you try to solve it by converting the fractions into decimals?

Fill the boxes in at least 2 different ways:

(a) $\square \times \square = 2.4$

(b) $\square \times \square = 14.5$

Find the following quotients given that $756 \div 36 = 21$:

(a) $75.6 \div 3.6$

(b) $7.56 \div 0.36$

(c) $756 \div 0.36$

(d) $75.6 \div 360$

(e) $7560 \div 3.6$

(f) $7.56 \div 0.36$

Find the missing cells if each cell represents a÷b:

Using the digits 2, 4, 5, 8, and 0 fill the boxes to get the:

(a) maximum product

(b) minimum product

(c) product greater than 150

(d) product nearest to 100

(e) product nearest to 5

Sort the following expressions in increasing order:

(a) $245.05 \times 0.942368$

(b) $245.05 \times 7.9682$

(c) $245.05 \div 7.9682$

(d) $245.05 \div 0.942368$

(e) $245.05$

(f) $7.9682$

Jonali and Pallabi play a game. Jonali says a fraction and Pallabi gives the equivalent decimal. Write Pallabi's answer in the blank spaces.

Jonali goes to the market to buy spices. She purchases 50 g of Cinnamon, 100 g of Cumin seeds, 25 g of Cardamom and 250 g of Pepper. Express each of the quantities in kilograms by writing them in terms of fractions as well as decimals.

Write the following fractions as a sum of fractions and also as decimals:

Can you give a simple rule to divide any number by a number of the form 1 followed by zeroes — 10, 100, 1000, etc.? For example, $\frac{123}{10}$, $\frac{24}{100}$ or $\frac{678}{1000}$? Look for a pattern in the previous problems.

Can the product of two decimals be a natural number?

Can the product of a decimal and a natural number be a natural number?

Observe the number of digits after the decimal point in the multiplier, the multiplicand and the product. Also note the number of zeroes in the denominator.

Suppose we know that 596 × 248 = 147808, can you immediately write down the product of 5.96 × 24.8?

Context: Observe the number of digits after the decimal point in the multiplier, the multiplicand and the product. Also note the number of zeroes in the denominator.

Q. By looking at the above examples, can you frame a rule to multiply two decimals?

When is the product of two decimals greater than both the numbers? When is it less than both the numbers?

What is 0.039 m in centimetres and millimetres?

Complete the table by dividing the decimals:

Can you find the quotients of $10 \div 9$, and $100 \div 11$?

Context: Let us consider the number 142857 that arose when dividing 1 by 7. Multiply 142857 by numbers from 1 to 6.

Q. What are the products? What do you notice?

Multiply 142857 by 7. What do you observe?

To find one such number, you can find $1 \div 17$ in decimal, and use the repeating block of digits.

Will the quotient be always greater than the dividend when the divisor is a decimal? Try it out with different values of the divisor.

Describe the relationship between the dividend, divisor, and the quotient. Create a table for capturing this relationship in different situations, like we did for multiplication.

What pattern do you observe? Why are 2 and 5 related in this way?

Do you know which month has this extra day?

With this new scheme of adding one extra day every 4th year, what is the number of days in 100 calendar years? Can you write an expression to calculate that number?

How many years are divisible by 4 in 100 years?

Can you form different expressions for the same question?

Context: The calendar designers decided that they will not add 1 extra day in every hundredth year.

Q. Can you write an expression for the number of days in 100 calendar years with this new adjustment?

With this final scheme of leap years can you calculate the number of calendar days in 10,000 years and the number of actual days the Earth will take to make 10,000 revolutions around the Sun? What is the difference? If there is a big difference, can you suggest a way to fix this problem?

Do you wonder how people figured out that the Earth completes one revolution around the Sun in exactly 364.2422 days?