A Peek Beyond the Point | IT

Question 15

Observe the given sequences of numbers. Identify the change after each term and extend the pattern:

(a) 44, 43104\frac{3}{10}, 46104\frac{6}{10}, ______, ______, ______

(b) 82108\frac{2}{10}, 87108\frac{7}{10}, 92109\frac{2}{10}, ______, ______, ______

(c) 76107\frac{6}{10}, 87108\frac{7}{10}, ______, ______, ______, ______

(d) 57105\frac{7}{10}, 53105\frac{3}{10}, ______, ______, ______, ______

(e) 1351013\frac{5}{10}, 1313, 1251012\frac{5}{10}, ______, ______, ______, ______

(f) 1151011\frac{5}{10}, 1041010\frac{4}{10}, 93109\frac{3}{10}, ______, ______, ______

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Solution

We will find the change between terms. Then we will extend each sequence.

Step 1 — Extending sequence (a)

Let us look at the first two numbers. The first number is 44. The second number is 43104\frac{3}{10}. We find the difference between them.

4310−44\frac{3}{10} - 4

=310= \frac{3}{10}

Now, let us look at the next two numbers. The second number is 43104\frac{3}{10}. The third number is 46104\frac{6}{10}. We find the difference between them.

4610−43104\frac{6}{10} - 4\frac{3}{10}

=310= \frac{3}{10}

The change is always adding 310\frac{3}{10}. Let us add 310\frac{3}{10} to find the next terms. The third term is 46104\frac{6}{10}. The fourth term is 4610+3104\frac{6}{10} + \frac{3}{10}.

4610+3104\frac{6}{10} + \frac{3}{10}

=4910= 4\frac{9}{10}

The fifth term is 4910+3104\frac{9}{10} + \frac{3}{10}.

4910+3104\frac{9}{10} + \frac{3}{10}

=41210= 4\frac{12}{10}

=4+1210= 4 + 1\frac{2}{10}

=5210= 5\frac{2}{10}

The sixth term is 5210+3105\frac{2}{10} + \frac{3}{10}.

5210+3105\frac{2}{10} + \frac{3}{10}

=5510= 5\frac{5}{10}

Pattern: Add 310\boxed{\text{Pattern: Add } \frac{3}{10}}

Step 2 — Extending sequence (b)

Let us look at the first two numbers. The first number is 82108\frac{2}{10}. The second number is 87108\frac{7}{10}. We find the difference between them.

8710−82108\frac{7}{10} - 8\frac{2}{10}

=510= \frac{5}{10}

Now, let us look at the next two numbers. The second number is 87108\frac{7}{10}. The third number is 92109\frac{2}{10}. We find the difference between them.

9210−87109\frac{2}{10} - 8\frac{7}{10}

=81210−8710= 8\frac{12}{10} - 8\frac{7}{10}

=510= \frac{5}{10}

The change is always adding 510\frac{5}{10}. Let us add 510\frac{5}{10} to find the next terms. The third term is 92109\frac{2}{10}. The fourth term is 9210+5109\frac{2}{10} + \frac{5}{10}.

9210+5109\frac{2}{10} + \frac{5}{10}

=9710= 9\frac{7}{10}

The fifth term is 9710+5109\frac{7}{10} + \frac{5}{10}.

9710+5109\frac{7}{10} + \frac{5}{10}

=91210= 9\frac{12}{10}

=10210= 10\frac{2}{10}

The sixth term is 10210+51010\frac{2}{10} + \frac{5}{10}.

10210+51010\frac{2}{10} + \frac{5}{10}

=10710= 10\frac{7}{10}

Pattern: Add 510\boxed{\text{Pattern: Add } \frac{5}{10}}

Step 3 — Extending sequence (c)

Let us look at the first two numbers. The first number is 76107\frac{6}{10}. The second number is 87108\frac{7}{10}. We find the difference between them.

8710−76108\frac{7}{10} - 7\frac{6}{10}

=(8−7)+(710−610)= (8-7) + (\frac{7}{10} - \frac{6}{10})

=1+110= 1 + \frac{1}{10}

=1110= 1\frac{1}{10}

The change is always adding 11101\frac{1}{10}. Let us add 11101\frac{1}{10} to find the next terms. The second term is 87108\frac{7}{10}. The third term is 8710+11108\frac{7}{10} + 1\frac{1}{10}.

8710+11108\frac{7}{10} + 1\frac{1}{10}

=(8+1)+(710+110)= (8+1) + (\frac{7}{10} + \frac{1}{10})

=9810= 9\frac{8}{10}

The fourth term is 9810+11109\frac{8}{10} + 1\frac{1}{10}.

9810+11109\frac{8}{10} + 1\frac{1}{10}

=(9+1)+(810+110)= (9+1) + (\frac{8}{10} + \frac{1}{10})

=10910= 10\frac{9}{10}

The fifth term is 10910+111010\frac{9}{10} + 1\frac{1}{10}.

10910+111010\frac{9}{10} + 1\frac{1}{10}

=(10+1)+(910+110)= (10+1) + (\frac{9}{10} + \frac{1}{10})

=11+1010= 11 + \frac{10}{10}

=11+1= 11 + 1

=12= 12

The sixth term is 12+111012 + 1\frac{1}{10}.

12+111012 + 1\frac{1}{10}

=13110= 13\frac{1}{10}

Pattern: Add 1110\boxed{\text{Pattern: Add } 1\frac{1}{10}}

Step 4 — Extending sequence (d)

Let us look at the first two numbers. The first number is 57105\frac{7}{10}. The second number is 53105\frac{3}{10}. The second number is smaller than the first. We find the amount subtracted.

5710−53105\frac{7}{10} - 5\frac{3}{10}

=410= \frac{4}{10}

The change is always subtracting 410\frac{4}{10}. Let us subtract 410\frac{4}{10} to find the next terms. The second term is 53105\frac{3}{10}. The third term is 5310−4105\frac{3}{10} - \frac{4}{10}.

5310−4105\frac{3}{10} - \frac{4}{10}

=41310−410= 4\frac{13}{10} - \frac{4}{10}

=4910= 4\frac{9}{10}

The fourth term is 4910−4104\frac{9}{10} - \frac{4}{10}.

4910−4104\frac{9}{10} - \frac{4}{10}

=4510= 4\frac{5}{10}

The fifth term is 4510−4104\frac{5}{10} - \frac{4}{10}.

4510−4104\frac{5}{10} - \frac{4}{10}

=4110= 4\frac{1}{10}

The sixth term is 4110−4104\frac{1}{10} - \frac{4}{10}.

4110−4104\frac{1}{10} - \frac{4}{10}

=31110−410= 3\frac{11}{10} - \frac{4}{10}

=3710= 3\frac{7}{10}

Pattern: Subtract 410\boxed{\text{Pattern: Subtract } \frac{4}{10}}

Step 5 — Extending sequence (e)

Let us look at the first two numbers. The first number is 1351013\frac{5}{10}. The second number is 1313. The second number is smaller than the first. We find the amount subtracted.

13510−1313\frac{5}{10} - 13

=510= \frac{5}{10}

Now, let us look at the next two numbers. The second number is 1313. The third number is 1251012\frac{5}{10}. We find the amount subtracted.

13−1251013 - 12\frac{5}{10}

=121010−12510= 12\frac{10}{10} - 12\frac{5}{10}

=510= \frac{5}{10}

The change is always subtracting 510\frac{5}{10}. Let us subtract 510\frac{5}{10} to find the next terms. The third term is 1251012\frac{5}{10}. The fourth term is 12510−51012\frac{5}{10} - \frac{5}{10}.

12510−51012\frac{5}{10} - \frac{5}{10}

=12= 12

The fifth term is 12−51012 - \frac{5}{10}.

12−51012 - \frac{5}{10}

=111010−510= 11\frac{10}{10} - \frac{5}{10}

=11510= 11\frac{5}{10}

The sixth term is 11510−51011\frac{5}{10} - \frac{5}{10}.

11510−51011\frac{5}{10} - \frac{5}{10}

=11= 11

The seventh term is 11−51011 - \frac{5}{10}.

11−51011 - \frac{5}{10}

=101010−510= 10\frac{10}{10} - \frac{5}{10}

=10510= 10\frac{5}{10}

Pattern: Subtract 510\boxed{\text{Pattern: Subtract } \frac{5}{10}}

Step 6 — Extending sequence (f)

Let us look at the first two numbers. The first number is 1151011\frac{5}{10}. The second number is 1041010\frac{4}{10}. The second number is smaller than the first. We find the amount subtracted.

11510−1041011\frac{5}{10} - 10\frac{4}{10}

=(11−10)+(510−410)= (11-10) + (\frac{5}{10} - \frac{4}{10})

=1+110= 1 + \frac{1}{10}

=1110= 1\frac{1}{10}

Now, let us look at the next two numbers. The second number is 1041010\frac{4}{10}. The third number is 93109\frac{3}{10}. We find the amount subtracted.

10410−931010\frac{4}{10} - 9\frac{3}{10}

=(10−9)+(410−310)= (10-9) + (\frac{4}{10} - \frac{3}{10})

=1+110= 1 + \frac{1}{10}

=1110= 1\frac{1}{10}

The change is always subtracting 11101\frac{1}{10}. Let us subtract 11101\frac{1}{10} to find the next terms. The third term is 93109\frac{3}{10}. The fourth term is 9310−11109\frac{3}{10} - 1\frac{1}{10}.

9310−11109\frac{3}{10} - 1\frac{1}{10}

=(9−1)+(310−110)= (9-1) + (\frac{3}{10} - \frac{1}{10})

=8210= 8\frac{2}{10}

The fifth term is 8210−11108\frac{2}{10} - 1\frac{1}{10}.

8210−11108\frac{2}{10} - 1\frac{1}{10}

=(8−1)+(210−110)= (8-1) + (\frac{2}{10} - \frac{1}{10})

=7110= 7\frac{1}{10}

The sixth term is 7110−11107\frac{1}{10} - 1\frac{1}{10}.

7110−11107\frac{1}{10} - 1\frac{1}{10}

=(7−1)+(110−110)= (7-1) + (\frac{1}{10} - \frac{1}{10})

=6= 6

Pattern: Subtract 1110\boxed{\text{Pattern: Subtract } 1\frac{1}{10}}

Answer

(a) 44, 43104\frac{3}{10}, 46104\frac{6}{10}, 4910\mathbf{4\frac{9}{10}}, 5210\mathbf{5\frac{2}{10}}, 5510\mathbf{5\frac{5}{10}} (b) 82108\frac{2}{10}, 87108\frac{7}{10}, 92109\frac{2}{10}, 9710\mathbf{9\frac{7}{10}}, 10210\mathbf{10\frac{2}{10}}, 10710\mathbf{10\frac{7}{10}} (c) 76107\frac{6}{10}, 87108\frac{7}{10}, 9810\mathbf{9\frac{8}{10}}, 10910\mathbf{10\frac{9}{10}}, 12\mathbf{12}, 13110\mathbf{13\frac{1}{10}} (d) 57105\frac{7}{10}, 53105\frac{3}{10}, 4910\mathbf{4\frac{9}{10}}, 4510\mathbf{4\frac{5}{10}}, 4110\mathbf{4\frac{1}{10}}, 3710\mathbf{3\frac{7}{10}} (e) 1351013\frac{5}{10}, 1313, 1251012\frac{5}{10}, 12\mathbf{12}, 11510\mathbf{11\frac{5}{10}}, 11\mathbf{11}, 10510\mathbf{10\frac{5}{10}} (f) 1151011\frac{5}{10}, 1041010\frac{4}{10}, 93109\frac{3}{10}, 8210\mathbf{8\frac{2}{10}}, 7110\mathbf{7\frac{1}{10}}, 6\mathbf{6}

More questions in IT

Q1

In the following figure, screws are placed above a scale. Measure them and write their length in the space provided.

Q2

Which scale helped you measure the length of the screws accurately? Why?

Q3

What is the meaning of 27102 \frac{7}{10} cm (the length of the first screw)?

Q4

Context: As seen on the ruler, the unit length between two consecutive numbers is divided into 10 equal parts. To get the length 2710 cm2\frac{7}{10}\text{ cm}, we go from 0 to 2 and then take seven parts of 110\frac{1}{10}. The length of the screw is 2 cm2\text{ cm} and 710 cm\frac{7}{10}\text{ cm}. Similarly, we can make sense of the length 3210 cm3\frac{2}{10}\text{ cm}.

Q. Can you explain why the unit was divided into smaller parts to measure the screws?

Q5

Measure the following objects using a scale and write their measurements in centimeters (as shown earlier for the lengths of the screws): pen, sharpener, and any other object of your choice.

Q6

Write the measurements of the objects shown in the picture:

Q7

For the objects shown below, write their lengths in two ways and read them aloud. An example is given for the USB cable. (Note that the unit length used in each diagram is not the same).

Q8

Arrange these lengths in increasing order:

(a) 910\frac{9}{10}

(b) 17101 \frac{7}{10}

(c) 13010\frac{130}{10}

(d) 1311013 \frac{1}{10}

(e) 1051010 \frac{5}{10}

(f) 76107 \frac{6}{10}

(g) 67106 \frac{7}{10}

(h) 410\frac{4}{10}

Q9

Arrange the following lengths in increasing order: 41104\frac{1}{10}, 410\frac{4}{10}, 4110\frac{41}{10}, 4111041\frac{1}{10}.

Q10

The lengths of the body parts of a honeybee are given. Find its total length.

Head: 23102 \frac{3}{10} units Thorax: 54105 \frac{4}{10} units Abdomen: 75107 \frac{5}{10} units

Q11

The length of Shylaja's hand is 1241012 \frac{4}{10} units, and her palm is 67106 \frac{7}{10} units, as shown in the picture. What is the length of the longest (middle) finger?

Q12

Context: The length of Shylaja's hand is 1241012 \frac{4}{10} units, and her palm is 67106 \frac{7}{10} units.

Q. Try computing the difference by converting both lengths to tenths.

Q13

A Celestial Pearl Danio's length is 24102 \frac{4}{10} cm, and the length of a Philippine Goby is 910\frac{9}{10} cm. What is the difference in their lengths?

Q14

How big are these fish compared to your finger?

Q15

Observe the given sequences of numbers. Identify the change after each term and extend the pattern:

(a) 44, 43104\frac{3}{10}, 46104\frac{6}{10}, ______, ______, ______

(b) 82108\frac{2}{10}, 87108\frac{7}{10}, 92109\frac{2}{10}, ______, ______, ______

(c) 76107\frac{6}{10}, 87108\frac{7}{10}, ______, ______, ______, ______

(d) 57105\frac{7}{10}, 53105\frac{3}{10}, ______, ______, ______, ______

(e) 1351013\frac{5}{10}, 1313, 1251012\frac{5}{10}, ______, ______, ______, ______

(f) 1151011\frac{5}{10}, 1041010\frac{4}{10}, 93109\frac{3}{10}, ______, ______, ______

Q16

What is the length of this smaller part? How many such smaller parts make a unit length?

Q17

How many one-hundredths make one-tenth? Can we also say that the length is 4 units and 45 one-hundredths?

Q18

Observe the figure below. Notice the markings and the corresponding lengths written in the boxes when measured from 0. Fill the lengths in the empty boxes.

Q19

For the lengths shown below write the measurements and read out the measures in words.

Q20

In each group, identify the longest and the shortest lengths. Mark each length on the scale.

(a) 310,3100,33100\frac{3}{10}, \frac{3}{100}, \frac{33}{100}

(b) 3110,3010,13103\frac{1}{10}, \frac{30}{10}, 1\frac{3}{10}

(c) 45100,54100,510,410\frac{45}{100}, \frac{54}{100}, \frac{5}{10}, \frac{4}{10}

(d) 3610,36100,361061003\frac{6}{10}, 3\frac{6}{100}, 3\frac{6}{10}\frac{6}{100}

(e) 810,2100,9100,18100\frac{8}{10}, \frac{2}{100}, \frac{9}{100}, 1\frac{8}{100}

(f) 73105100,7510,7411007\frac{3}{10}\frac{5}{100}, 7\frac{5}{10}, 7\frac{41}{100}

(g) 651015100,587100,57100\frac{65}{10}\frac{15}{100}, 5\frac{87}{100}, 5\frac{7}{100}

Q21

Context: What will be the sum of 15 310 410015\ \frac{3}{10}\ \frac{4}{100} and 2 610 81002\ \frac{6}{10}\ \frac{8}{100}? This can be solved in different ways. Some are shown below.

Are both these methods different?

Q22

Observe the addition done below for 483+268483 + 268. Do you see any similarities between the methods shown above?

(400+80+3)+(200+60+8)(400 + 80 + 3) + (200 + 60 + 8)

=(400+200)+(80+60)+(3+8)= (400 + 200) + (80 + 60) + (3 + 8)

=600+140+11= 600 + 140 + 11

=600+150+1= 600 + 150 + 1

=700+50+1= 700 + 50 + 1

=751= 751

Q25

Can we extend this further?

Q26

We can ask similar questions about fractional parts:

(a) How many thousandths make one unit? (b) How many thousandths make one tenth? (c) How many thousandths make one hundredth? (d) How many tenths make one ten? (e) How many hundredths make one ten?

Q27

Make a few more questions of this kind and answer them.

Q28

Can the quantity 42104 \frac{2}{10} be written as 42 (skipping the 110\frac{1}{10} in 2×1102 \times \frac{1}{10})?

If yes, how would we know if 42 means 4 tens and 2 units or it means 4 units and 2 tenths?

Q29

Make a place value table similar to the one above. Write each quantity in decimal form and in terms of place value, and read the number:

(a) 2 ones, 3 tenths and 5 hundredths

(b) 1 ten and 5 tenths

(c) 4 ones and 6 hundredths

(d) 1 hundred, 1 one and 1 hundredth

(e) 8100\frac{8}{100} and 910\frac{9}{10}

(f) 5100\frac{5}{100}

(g) 110\frac{1}{10}

(h) 211002\frac{1}{100}, 41104\frac{1}{10} and 7710007\frac{7}{1000}

Q30

Write these quantities in decimal form: (a) 234 hundredths (b) 105 tenths

Q31

Fill in the blanks below (mm <-> cm)

Q32

Fill in the blanks below (cm <-> m):

Q33

How many mm does 1 meter have?

Q34

Can we write 1 mm=11000 m1\text{ mm} = \frac{1}{1000}\text{ m}?

Q35

Fill in the blanks below (g <-> kg)

Q36

Fill in the blanks below (rupee <-> paise)

Q37

Name all the divisions between 1 and 1.1 on the number line.

Q38

Identify and write the decimal numbers against the letters.

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