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A Peek Beyond the Point

CBSE · Class 7 · Mathematics

NCERT Solutions for A Peek Beyond the Point — CBSE Class 7 Mathematics.

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61 Questions Solved · 7 Sections

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Solve This — Difference Using Hundredths

1What is the difference 153104100−2610810015 \frac{3}{10} \frac{4}{100} - 2 \frac{6}{10} \frac{8}{100}?Show solution
Given: 153104100−2610810015 \frac{3}{10} \frac{4}{100} - 2 \frac{6}{10} \frac{8}{100}

Step 1: Convert both numbers to hundredths notation (decimal form).

15310410015 \frac{3}{10} \frac{4}{100} means 15+310+4100=15+30100+4100=15+34100=15.3415 + \frac{3}{10} + \frac{4}{100} = 15 + \frac{30}{100} + \frac{4}{100} = 15 + \frac{34}{100} = 15.34

261081002 \frac{6}{10} \frac{8}{100} means 2+610+8100=2+60100+8100=2+68100=2.682 + \frac{6}{10} + \frac{8}{100} = 2 + \frac{60}{100} + \frac{8}{100} = 2 + \frac{68}{100} = 2.68

Step 2: Subtract.

15.34−2.6815.34 - 2.68

Breaking it down by place value:
(10+5+310+4100)−(2+610+8100)(10 + 5 + \tfrac{3}{10} + \tfrac{4}{100}) - (2 + \tfrac{6}{10} + \tfrac{8}{100})

=(15−2)+(310−610)+(4100−8100)= (15 - 2) + (\tfrac{3}{10} - \tfrac{6}{10}) + (\tfrac{4}{100} - \tfrac{8}{100})

Since 310<610\frac{3}{10} < \frac{6}{10}, regroup: borrow 1 unit from 15, converting it to 1010\frac{10}{10}:

=(14−2)+(1310−610)+(4100−8100)= (14 - 2) + (\tfrac{13}{10} - \tfrac{6}{10}) + (\tfrac{4}{100} - \tfrac{8}{100})

Since 4100<8100\frac{4}{100} < \frac{8}{100}, regroup: borrow 110\frac{1}{10} from 1310\frac{13}{10}, converting it to 10100\frac{10}{100}:

=(14−2)+(1210−610)+(14100−8100)= (14 - 2) + (\tfrac{12}{10} - \tfrac{6}{10}) + (\tfrac{14}{100} - \tfrac{8}{100})

=12+610+6100= 12 + \tfrac{6}{10} + \tfrac{6}{100}

=12.66= 12.66

**Answer: 15.34−2.68=12.6615.34 - 2.68 = 12.66**

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Figure it Out — Sums and Differences (Tenths and Hundredths Notation)

aFind: 310+34100\frac{3}{10} + 3\frac{4}{100}Show solution
Convert to decimal form:

310=0.30\frac{3}{10} = 0.30

34100=3+4100=3.043\frac{4}{100} = 3 + \frac{4}{100} = 3.04

Add:
0.30+3.04=3.340.30 + 3.04 = 3.34

**Answer: 3.343.34**

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bFind: 95107100+211031009\frac{5}{10}\frac{7}{100} + 2\frac{1}{10}\frac{3}{100}Show solution
Convert to decimal form:

95107100=9+510+7100=9.579\frac{5}{10}\frac{7}{100} = 9 + \frac{5}{10} + \frac{7}{100} = 9.57

21103100=2+110+3100=2.132\frac{1}{10}\frac{3}{100} = 2 + \frac{1}{10} + \frac{3}{100} = 2.13

Add:
9.57+2.13=11.709.57 + 2.13 = 11.70

**Answer: 11.7011.70**

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cFind: 156104100+14310610015\frac{6}{10}\frac{4}{100} + 14\frac{3}{10}\frac{6}{100}Show solution
Convert to decimal form:

156104100=15+610+4100=15.6415\frac{6}{10}\frac{4}{100} = 15 + \frac{6}{10} + \frac{4}{100} = 15.64

143106100=14+310+6100=14.3614\frac{3}{10}\frac{6}{100} = 14 + \frac{3}{10} + \frac{6}{100} = 14.36

Add:
15.64+14.36=30.0015.64 + 14.36 = 30.00

**Answer: 30.0030.00**

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dFind: 77100−441007\frac{7}{100} - 4\frac{4}{100}Show solution
Convert to decimal form:

77100=7+7100=7.077\frac{7}{100} = 7 + \frac{7}{100} = 7.07

44100=4+4100=4.044\frac{4}{100} = 4 + \frac{4}{100} = 4.04

Subtract:
7.07−4.04=3.037.07 - 4.04 = 3.03

**Answer: 3.033.03**

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eFind: 86100−531008\frac{6}{100} - 5\frac{3}{100}Show solution
Convert to decimal form:

86100=8+6100=8.068\frac{6}{100} = 8 + \frac{6}{100} = 8.06

53100=5+3100=5.035\frac{3}{100} = 5 + \frac{3}{100} = 5.03

Subtract:
8.06−5.03=3.038.06 - 5.03 = 3.03

**Answer: 3.033.03**

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fFind: 126102100−910910012\frac{6}{10}\frac{2}{100} - \frac{9}{10}\frac{9}{100}Show solution
Convert to decimal form:

126102100=12+610+2100=12.6212\frac{6}{10}\frac{2}{100} = 12 + \frac{6}{10} + \frac{2}{100} = 12.62

9109100=910+9100=0.99\frac{9}{10}\frac{9}{100} = \frac{9}{10} + \frac{9}{100} = 0.99

Subtract:
12.62−0.9912.62 - 0.99

Regrouping:
=12+0.62−0.99= 12 + 0.62 - 0.99
=11+1.62−0.99= 11 + 1.62 - 0.99
=11+0.63= 11 + 0.63
=11.63= 11.63

**Answer: 11.6311.63**

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Write the Detailed Place Value Computation for 84.691 − 77.345

1Write the detailed place value computation for 84.691−77.34584.691 - 77.345, and its compact form.Show solution
Detailed place value computation:

84.691−77.34584.691 - 77.345

=(80+4+0.6+0.09+0.001)−(70+7+0.3+0.04+0.005)= (80 + 4 + 0.6 + 0.09 + 0.001) - (70 + 7 + 0.3 + 0.04 + 0.005)

Ones place: 4−74 - 7: need to regroup. Borrow 10 from tens.
Tens place: 80→7080 \to 70, ones become 14−7=714 - 7 = 7

Tenths place: 0.6−0.3=0.30.6 - 0.3 = 0.3

Hundredths place: 0.09−0.04=0.050.09 - 0.04 = 0.05

Thousandths place: 0.001−0.0050.001 - 0.005: need to regroup. Borrow 0.010.01 from hundredths.
0.05→0.040.05 \to 0.04, thousandths become 0.011−0.005=0.0060.011 - 0.005 = 0.006

Tens place: 70−70=070 - 70 = 0

Putting it all together:
=(70−70)+(14−7)+(0.6−0.3)+(0.04−0.04)+(0.011−0.005)= (70 - 70) + (14 - 7) + (0.6 - 0.3) + (0.04 - 0.04) + (0.011 - 0.005)
=0+7+0.3+0.04+0.006= 0 + 7 + 0.3 + 0.04 + 0.006
=7.346= 7.346

Compact (column) form:
84.691−  77.3457.346\begin{array}{r} 84.691 \\ -\;77.345 \\ \hline 7.346 \end{array}

**Answer: 84.691−77.345=7.34684.691 - 77.345 = 7.346**

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Figure it Out — Section 3.4: Find the Sums

1aFind the sum: 5.3+2.65.3 + 2.6Show solution
Align decimal points and add:
5.3+  2.67.9\begin{array}{r} 5.3 \\ +\;2.6 \\ \hline 7.9 \end{array}

**Answer: 7.97.9**

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1bFind the sum: 18+8.818 + 8.8Show solution
**Write 1818 as 18.018.0 and add:**
18.0+  8.826.8\begin{array}{r} 18.0 \\ +\;8.8 \\ \hline 26.8 \end{array}

**Answer: 26.826.8**

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1cFind the sum: 2.15+5.262.15 + 5.26Show solution
2.15+  5.267.41\begin{array}{r} 2.15 \\ +\;5.26 \\ \hline 7.41 \end{array}

**Answer: 7.417.41**

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1dFind the sum: 9.01+9.109.01 + 9.10Show solution
9.01+  9.1018.11\begin{array}{r} 9.01 \\ +\;9.10 \\ \hline 18.11 \end{array}

**Answer: 18.1118.11**

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1eFind the sum: 29.19+9.9129.19 + 9.91Show solution
29.19+  9.9139.10\begin{array}{r} 29.19 \\ +\;9.91 \\ \hline 39.10 \end{array}

Tenths: 1+9=101 + 9 = 10, write 0 carry 1. Hundredths: 9+1=109 + 1 = 10, write 0 carry 1 (already counted). Let me redo carefully:

Hundredths: 9+1=109 + 1 = 10, write 0, carry 1.
Tenths: 1+9+1=111 + 9 + 1 = 11, write 1, carry 1.
Ones: 9+9+1=199 + 9 + 1 = 19, write 9, carry 1.
Tens: 2+0+1=32 + 0 + 1 = 3.

=39.10= 39.10

**Answer: 39.1039.10**

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1fFind the sum: 0.934+0.60.934 + 0.6Show solution
**Write 0.60.6 as 0.6000.600:**
0.934+  0.6001.534\begin{array}{r} 0.934 \\ +\;0.600 \\ \hline 1.534 \end{array}

**Answer: 1.5341.534**

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1gFind the sum: 0.75+0.030.75 + 0.03Show solution
0.75+  0.030.78\begin{array}{r} 0.75 \\ +\;0.03 \\ \hline 0.78 \end{array}

**Answer: 0.780.78**

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1hFind the sum: 6.236+0.4876.236 + 0.487Show solution
6.236+  0.4876.723\begin{array}{r} 6.236 \\ +\;0.487 \\ \hline 6.723 \end{array}

Thousandths: 6+7=136 + 7 = 13, write 3, carry 1.
Hundredths: 3+8+1=123 + 8 + 1 = 12, write 2, carry 1.
Tenths: 2+4+1=72 + 4 + 1 = 7.
Ones: 6+0=66 + 0 = 6.

**Answer: 6.7236.723**

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Figure it Out — Section 3.4: Find the Differences

2aFind the difference: 5.6−2.35.6 - 2.3Show solution
5.6−  2.33.3\begin{array}{r} 5.6 \\ -\;2.3 \\ \hline 3.3 \end{array}

**Answer: 3.33.3**

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2bFind the difference: 18−8.818 - 8.8Show solution
**Write 1818 as 18.018.0:**
18.0−  8.89.2\begin{array}{r} 18.0 \\ -\;8.8 \\ \hline 9.2 \end{array}

Tenths: 0−80 - 8, regroup: borrow 1 from ones. 10−8=210 - 8 = 2.
Ones: 8−1−88 - 1 - 8, regroup: borrow from tens. 18−1−8=918 - 1 - 8 = 9.
Tens: 1−0=11 - 0 = 1... wait: 18.0−8.818.0 - 8.8:
Ones: 8−8=08 - 8 = 0 after borrowing. Tens: 1−0=11 - 0 = 1...

Let me redo: 18.0−8.818.0 - 8.8
Tenths: 0−80 - 8, borrow 1 tenth-group from ones: ones become 7, tenths become 10. 10−8=210 - 8 = 2.
Ones: 7−87 - 8, borrow from tens: tens become 0 (from 1), ones become 17. 17−8=917 - 8 = 9.
Tens: 0−0=00 - 0 = 0.

Result: 9.29.2

**Answer: 9.29.2**

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2cFind the difference: 10.4−4.510.4 - 4.5Show solution
Tenths: 4−54 - 5, borrow: ones become 9, tenths become 14. 14−5=914 - 5 = 9.
Ones: 9−4=59 - 4 = 5... wait: 10.4−4.510.4 - 4.5
Ones digit of 10 is 0, tens digit is 1.
Tenths: 4−54 - 5, borrow from ones. Ones: 0→0 \to need to borrow from tens. Tens: 1→01 \to 0, ones become 10. Now borrow from ones for tenths: ones 10→910 \to 9, tenths 4+10=144 + 10 = 14. 14−5=914 - 5 = 9.
Ones: 9−4=59 - 4 = 5.
Tens: 0−0=00 - 0 = 0.

Result: 5.95.9

**Answer: 5.95.9**

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2dFind the difference: 17−16.19817 - 16.198Show solution
**Write 1717 as 17.00017.000:**

17.000−16.19817.000 - 16.198

Thousandths: 0−80 - 8, borrow chain: 10−8=210 - 8 = 2.
Hundredths: 0−1−90 - 1 - 9 (after borrowing): regroup again: 10−1−9=010 - 1 - 9 = 0...

Let me do it step by step:
Thousandths: 0−80 - 8, borrow from hundredths. Hundredths become −1-1 (need to borrow too).
Hundredths: 0−1−90 - 1 - 9, borrow from tenths. Tenths become −1-1.
Tenths: 0−1−10 - 1 - 1, borrow from ones. Ones become −1-1.
Ones: 7−1−87 - 1 - 8, borrow from tens. Tens become 00.

More cleanly: 17.000−16.19817.000 - 16.198
=17.000−16.198=0.802= 17.000 - 16.198 = 0.802

Verification: 16.198+0.802=17.00016.198 + 0.802 = 17.000 ✓

**Answer: 0.8020.802**

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2eFind the difference: 17−0.0517 - 0.05Show solution
**Write 1717 as 17.0017.00:**
17.00−0.0517.00 - 0.05

Hundredths: 0−50 - 5, borrow. Tenths: 0−0−10 - 0 - 1 (after borrow), borrow from ones. Ones: 7→67 \to 6, tenths →10\to 10, then tenths 10−1=910 - 1 = 9, hundredths 10−5=510 - 5 = 5.
Ones: 6−0=66 - 0 = 6. Tens: 1−0=11 - 0 = 1.

Result: 16.9516.95

**Answer: 16.9516.95**

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2fFind the difference: 34.505−18.134.505 - 18.1Show solution
**Write 18.118.1 as 18.10018.100:**
34.505−18.10034.505 - 18.100

Thousandths: 5−0=55 - 0 = 5.
Hundredths: 0−0=00 - 0 = 0.
Tenths: 5−1=45 - 1 = 4.
Ones: 4−84 - 8, borrow from tens: tens 3→23 \to 2, ones 14−8=614 - 8 = 6.
Tens: 2−1=12 - 1 = 1.

Result: 16.40516.405

**Answer: 16.40516.405**

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2gFind the difference: 9.9−9.099.9 - 9.09Show solution
**Write 9.99.9 as 9.909.90:**
9.90−9.099.90 - 9.09

Hundredths: 0−90 - 9, borrow from tenths. Tenths: 9→89 \to 8, hundredths 10−9=110 - 9 = 1.
Tenths: 8−0=88 - 0 = 8.
Ones: 9−9=09 - 9 = 0.

Result: 0.810.81

**Answer: 0.810.81**

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2hFind the difference: 6.236−0.4876.236 - 0.487Show solution
6.236−0.4876.236 - 0.487

Thousandths: 6−76 - 7, borrow. Hundredths: 3→23 \to 2, thousandths 16−7=916 - 7 = 9.
Hundredths: 2−82 - 8, borrow. Tenths: 2→12 \to 1, hundredths 12−8=412 - 8 = 4.
Tenths: 1−41 - 4, borrow. Ones: 6→56 \to 5, tenths 11−4=711 - 4 = 7.
Ones: 5−0=55 - 0 = 5.

Result: 5.7495.749

**Answer: 5.7495.749**

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Decimal Sequences

1Continue the sequence 4.4,4.8,5.2,5.6,6.0,…4.4, 4.8, 5.2, 5.6, 6.0, \ldots and write the next 3 terms.Show solution
Pattern identified: Each term increases by 0.40.4.

6.0+0.4=6.46.0 + 0.4 = 6.4
6.4+0.4=6.86.4 + 0.4 = 6.8
6.8+0.4=7.26.8 + 0.4 = 7.2

**The next 3 terms are: 6.4,6.8,7.26.4, 6.8, 7.2**

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Figure it Out — Decimal Place Value (Fractions and Decimals)

1aConvert 5100\frac{5}{100} into decimal.Show solution
5100=0.05\frac{5}{100} = 0.05

**Answer: 0.050.05**

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1bConvert 161000\frac{16}{1000} into decimal.Show solution
161000=0.016\frac{16}{1000} = 0.016

**Answer: 0.0160.016**

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1cConvert 1210\frac{12}{10} into decimal.Show solution
1210=1.2\frac{12}{10} = 1.2

**Answer: 1.21.2**

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1dConvert 2541000\frac{254}{1000} into decimal.Show solution
2541000=0.254\frac{254}{1000} = 0.254

**Answer: 0.2540.254**

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2aConvert 0.340.34 into a sum of tenths, hundredths and thousandths.Show solution
0.34=310+4100+010000.34 = \frac{3}{10} + \frac{4}{100} + \frac{0}{1000}

Or simply: 0.34=30.34 = 3 tenths +4+ 4 hundredths +0+ 0 thousandths.

**Answer: 0.34=310+41000.34 = \frac{3}{10} + \frac{4}{100}**

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2bConvert 1.021.02 into a sum of tenths, hundredths and thousandths.Show solution
1.02=1+010+21001.02 = 1 + \frac{0}{10} + \frac{2}{100}

=1= 1 one +0+ 0 tenths +2+ 2 hundredths +0+ 0 thousandths.

**Answer: 1.02=1+21001.02 = 1 + \frac{2}{100}**

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2cConvert 0.80.8 into a sum of tenths, hundredths and thousandths.
2dConvert 0.3620.362 into a sum of tenths, hundredths and thousandths.
3What decimal number does each letter represent in the number line? (Number line image not visible — general method stated.)
4aArrange in descending order: 11.01,1.011,1.101,11.10,1.0111.01, 1.011, 1.101, 11.10, 1.01
4bArrange in descending order: 2.567,2.675,2.768,2.499,2.6982.567, 2.675, 2.768, 2.499, 2.698
4cArrange in descending order: 4.678 g,4.595 g,4.600 g,4.656 g,4.666 g4.678\text{ g}, 4.595\text{ g}, 4.600\text{ g}, 4.656\text{ g}, 4.666\text{ g}
4dArrange in descending order: 33.13 m,33.31 m,33.133 m,33.331 m,33.313 m33.13\text{ m}, 33.31\text{ m}, 33.133\text{ m}, 33.331\text{ m}, 33.313\text{ m}
5aUsing the digits 1,4,0,8,61, 4, 0, 8, 6 make the decimal number closest to 3030.
5bUsing the digits 1,4,0,8,61, 4, 0, 8, 6 make the smallest possible decimal number between 100100 and 10001000.
6Will a decimal number with more digits be greater than a decimal number with fewer digits?
7Mahi purchases 0.25 kg0.25\text{ kg} of beans, 0.3 kg0.3\text{ kg} of carrots, 0.5 kg0.5\text{ kg} of potatoes, 0.2 kg0.2\text{ kg} of capsicums, and 0.05 kg0.05\text{ kg} of ginger. Calculate the total weight.
8Pinto supplies 3.793.79 L, 4.24.2 L, and 4.254.25 L of milk in the first three days. In 6 days, he supplies 2525 litres. Find the total quantity supplied in the last three days.
9Tinku weighed 35.7535.75 kg in January and 34.5034.50 kg in February. Has he gained or lost weight? How much is the change?
10Extend the pattern: 5.5,6.4,6.39,7.29,7.28,6.18,6.17,_,_5.5, 6.4, 6.39, 7.29, 7.28, 6.18, 6.17, \_, \_
11How many millimeters make 1 kilometer?
12Indian Railways offers optional travel insurance for passengers at 45 paise per passenger. If 1 lakh people opt for insurance in a day, what is the total insurance fee paid?
13aWhich is greater: 101000\frac{10}{1000} or 110\frac{1}{10}?
13bWhich is greater: One-hundredth or 90 thousandths?
13cWhich is greater: One-thousandth or 90 hundredths?
14aWrite the decimal form: 87 ones, 5 tenths and 60 hundredths (example given as =88.10= 88.10).
14bWrite the decimal form: 12 tens and 12 tenths.
14cWrite the decimal form: 10 tens, 10 ones, 10 tenths, and 10 hundredths.
14dWrite the decimal form: 25 tens, 25 ones, 25 tenths, and 25 hundredths.
15Using each digit 00–99 not more than once, fill the boxes so that the sum is closest to 10.510.5. (Box image not visible — general approach given.)
16aWrite 12\frac{1}{2} in decimal form.
16bWrite 32\frac{3}{2} in decimal form.
16cWrite 14\frac{1}{4} in decimal form.
16dWrite 34\frac{3}{4} in decimal form.
16eWrite 15\frac{1}{5} in decimal form.
16fWrite 45\frac{4}{5} in decimal form.

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