Another Peek Beyond the Point

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CLASS VII Mathematics ~6 marks/year Ch 12 of 15
Another Peek Beyond the Point

Class 7 · Mathematics · NCERT chapter notes · Akanksha Classes

Snapshot
  • This chapter takes a deeper look at decimals — numbers with a part "beyond the point", such as $3.7$ or $0.045$.
  • Each place after the decimal point has a value ten times smaller than the one before: tenths, hundredths, thousandths.
  • Decimals and fractions are two faces of the same number: $0.25=\dfrac{25}{100}=\dfrac14$.
  • We can compare decimals place by place, and add, subtract, multiply and divide them with care about the point's position.
  • Multiplying by $10,100,1000$ shifts the point right; dividing shifts it left.
  • Decimals run our money, measurements, and digital world — every price tag and ruler uses them.
  • Weightage: ~6 marks/year — place value, conversions, operations, and measurement word problems.
Detailed Notes

1. What lies beyond the point

A decimal number has two parts separated by a decimal point: the whole-number part on the left and the fractional part on the right. In $47.382$, the part $47$ is whole and $.382$ is the fraction "beyond the point". This chapter explores that fractional part in detail.

Each place to the right of the point is worth one-tenth of the place to its left:

First place = tenths $\left(\dfrac{1}{10}\right)$, second = hundredths $\left(\dfrac{1}{100}\right)$, third = thousandths $\left(\dfrac{1}{1000}\right)$, and so on.
Worked example — expanded form of $47.382$

$47.382=40+7+\dfrac{3}{10}+\dfrac{8}{100}+\dfrac{2}{1000}$. The digit $3$ stands for three tenths, $8$ for eight hundredths, and $2$ for two thousandths.

2. Place value table

Tens Ones · Tenths Hundredths Thousandths
47·382

Reading $47.382$: "forty-seven point three eight two." Note we read the digits after the point one by one, not as "three hundred eighty-two".

3. Decimals as fractions, and fractions as decimals

A decimal is just a fraction whose denominator is $10,100,1000,\dots$

$0.7=\dfrac{7}{10}$,   $0.45=\dfrac{45}{100}$,   $0.125=\dfrac{125}{1000}=\dfrac18$.

To write a fraction as a decimal, make the denominator a power of $10$ (when possible) or divide the numerator by the denominator.

Worked example — $\dfrac{3}{4}$ as a decimal

Multiply top and bottom by $25$: $\dfrac{3}{4}=\dfrac{3\times25}{4\times25}=\dfrac{75}{100}=0.75$.

Worked example — $0.36$ as a fraction in lowest terms

$0.36=\dfrac{36}{100}$. Divide top and bottom by their HCF $4$: $\dfrac{36}{100}=\dfrac{9}{25}$.

4. Comparing and ordering decimals

To compare decimals, line up the decimal points and compare digit by digit from the left.

  • First compare whole-number parts; the bigger whole part wins.
  • If those are equal, compare tenths, then hundredths, and so on.
  • Adding zeros at the end of the decimal part does not change the value: $0.5=0.50=0.500$. This makes comparing easier.
Worked example — order $0.7,\ 0.65,\ 0.709$

Write them with equal places: $0.700,\ 0.650,\ 0.709$. Compare: $0.650<0.700<0.709$. So in increasing order: $0.65<0.7<0.709$.

5. Adding and subtracting decimals

The golden rule is to line up the decimal points so that tenths sit under tenths, hundredths under hundredths, and so on. Fill empty places with zeros, then add or subtract as with whole numbers and bring the point straight down.

Always align the decimal points vertically before adding or subtracting.
Worked example — add $12.5+3.74+0.6$

Write as $12.50,\ 03.74,\ 00.60$ aligned by the point. Adding: $12.50+3.74=16.24$; then $16.24+0.60=16.84$. So the sum is $16.84$.

Worked example — subtract $25.3-8.47$

Write $25.30-8.47$. Subtract: $25.30-8.47=16.83$.

6. Multiplying decimals

By 10, 100, 1000

Multiplying by a power of $10$ simply shifts the decimal point to the right by as many places as there are zeros.

$\times10\Rightarrow$ point moves $1$ place right; $\times100\Rightarrow2$ places; $\times1000\Rightarrow3$ places.
Worked example

$4.56\times10=45.6$; $4.56\times100=456$; $0.072\times1000=72$.

By another decimal

Ignore the points and multiply as whole numbers. Then count the total number of decimal places in the two factors and put the point that many places from the right in the answer.

Worked example — $1.2\times0.05$

Multiply $12\times5=60$. Decimal places: $1$ (in $1.2$) $+$ $2$ (in $0.05$) $=3$. So place the point $3$ from the right: $0.060=0.06$.

Worked example — $3.4\times2.5$

$34\times25=850$. Decimal places: $1+1=2$. So $3.4\times2.5=8.50=8.5$.

7. Dividing decimals

By 10, 100, 1000

Dividing by a power of $10$ shifts the decimal point to the left by as many places as there are zeros.

$\div10\Rightarrow$ point moves $1$ place left; $\div100\Rightarrow2$ places; $\div1000\Rightarrow3$ places.
Worked example

$45.6\div10=4.56$; $456\div100=4.56$; $7.2\div1000=0.0072$.

By a whole number

Worked example — $19.2\div4$

Divide as usual, keeping the point in line: $19.2\div4=4.8$.

By a decimal

Convert the divisor into a whole number by multiplying both numbers by the same power of $10$, then divide.

Worked example — $4.5\div0.5$

Multiply both by $10$: $45\div5=9$. So $4.5\div0.5=9$.

Worked example — $0.144\div0.12$

Multiply both by $100$: $14.4\div12=1.2$. So $0.144\div0.12=1.2$.

8. Decimals in measurement and money

Decimals make it easy to mix units. Remember the conversions:

$1$ cm $=0.01$ m,   $1$ mm $=0.1$ cm $=0.001$ m,   $1$ g $=0.001$ kg,   $1$ paisa $=$ ₹$0.01$.
Worked example — total length

A tailor uses $1.25$ m, $0.8$ m and $2.4$ m of cloth. Total $=1.25+0.80+2.40=4.45$ m.

Worked example — money change

A book costs ₹$148.50$ and you pay ₹$200$. Change $=200.00-148.50=$ ₹$51.50$.

Worked example — cost from rate

If $1$ kg of apples costs ₹$96.50$, then $2.5$ kg cost $96.50\times2.5$. $9650\times25=241250$; with $2+1=3$ decimal places, $=241.250=$ ₹$241.25$.

9. Rounding (estimating) decimals

Often we do not need the exact value — a sensible estimate is enough, especially for money and measurement. To round a decimal to a certain place, look at the digit just to the right of that place:

  • If it is $5$ or more, round the chosen place up by one.
  • If it is less than $5$, leave the chosen place unchanged (round down).
  • Drop all the digits after the chosen place.
Look at the next digit: $5$ or more rounds up, less than $5$ stays the same.
Worked example — round $7.846$

To the nearest tenth: the hundredths digit is $4$ (less than $5$), so $7.846\approx7.8$. To the nearest hundredth: the thousandths digit is $6$ (more than $5$), so $7.846\approx7.85$. To the nearest whole number: the tenths digit is $8$ (more than $5$), so $7.846\approx8$.

Worked example — estimating a bill

A shopper buys items costing ₹$48.75$, ₹$19.40$ and ₹$31.10$. Rounding each to the nearest rupee gives $49+19+31=$ ₹$99$, a quick estimate of the total (the exact total is ₹$99.25$).

10. Common mistakes to avoid

  • Not lining up decimal points when adding or subtracting.
  • Forgetting to count all decimal places in a product — e.g. $1.2\times0.05$ needs $3$ places.
  • Reading $0.382$ as "three hundred eighty-two" instead of "three eight two".
  • Thinking $0.5<0.45$ because "$45$ is bigger than $5$" — compare place value: $0.50>0.45$.
  • Moving the decimal point the wrong way: $\times10$ moves it right, $\div10$ moves it left.
  • Dropping a needed zero — $0.06$ is not the same as $0.6$.

11. Quick revision checklist

  • Places after the point: tenths, hundredths, thousandths (each $\tfrac{1}{10}$ of the one before).
  • Decimal $\leftrightarrow$ fraction: denominator is a power of $10$, then simplify.
  • Compare by aligning points and adding trailing zeros.
  • Add/subtract: line up the points.
  • Multiply: multiply as whole numbers, then count total decimal places.
  • $\times10,100,1000\Rightarrow$ point right; $\div10,100,1000\Rightarrow$ point left.
  • Divide by a decimal: make the divisor whole first (multiply both by a power of $10$).
  • Rounding: look at the next digit — $5$ or more rounds up, less rounds down.
Practice MCQs
1. The place value of $3$ in $47.382$ is:
  1. $3$
  2. $\dfrac{3}{10}$
  3. $\dfrac{3}{100}$
  4. $\dfrac{3}{1000}$
Answer: (B) The first place after the point is tenths, so $3$ means $\dfrac{3}{10}$.
2. $\dfrac{3}{4}$ written as a decimal is:
  1. $0.34$
  2. $0.75$
  3. $0.43$
  4. $0.075$
Answer: (B) $\dfrac34=\dfrac{75}{100}=0.75$.
3. $0.36$ as a fraction in lowest terms is:
  1. $\dfrac{36}{100}$
  2. $\dfrac{18}{50}$
  3. $\dfrac{9}{25}$
  4. $\dfrac{3}{8}$
Answer: (C) $\dfrac{36}{100}=\dfrac{9}{25}$ after dividing by $4$.
4. Which is the greatest?
  1. $0.7$
  2. $0.65$
  3. $0.709$
  4. $0.69$
Answer: (C) Writing equal places, $0.709$ is the largest.
5. $12.5+3.74+0.6$ equals:
  1. $16.84$
  2. $15.84$
  3. $16.04$
  4. $22.0$
Answer: (A) Aligning points: $12.50+3.74+0.60=16.84$.
6. $25.3-8.47$ equals:
  1. $17.83$
  2. $16.83$
  3. $16.93$
  4. $17.17$
Answer: (B) $25.30-8.47=16.83$.
7. $4.56\times100$ equals:
  1. $45.6$
  2. $456$
  3. $4560$
  4. $0.456$
Answer: (B) Multiplying by $100$ moves the point $2$ places right.
8. $1.2\times0.05$ equals:
  1. $0.6$
  2. $0.06$
  3. $0.006$
  4. $6.0$
Answer: (B) $12\times5=60$, with $1+2=3$ decimal places: $0.060=0.06$.
9. $7.2\div1000$ equals:
  1. $0.72$
  2. $0.072$
  3. $0.0072$
  4. $0.00072$
Answer: (C) Dividing by $1000$ moves the point $3$ places left.
10. $4.5\div0.5$ equals:
  1. $0.9$
  2. $9$
  3. $90$
  4. $0.09$
Answer: (B) Multiply both by $10$: $45\div5=9$.
11. The expanded form $40+7+\dfrac{3}{10}+\dfrac{8}{100}+\dfrac{2}{1000}$ equals:
  1. $47.382$
  2. $47.823$
  3. $4.7382$
  4. $473.82$
Answer: (A) The digits give $47.382$.
12. Which equals $0.5$?
  1. $0.05$
  2. $0.500$
  3. $5.0$
  4. $0.005$
Answer: (B) Trailing zeros do not change value: $0.5=0.500$.
13. $0.144\div0.12$ equals:
  1. $0.12$
  2. $1.2$
  3. $12$
  4. $0.012$
Answer: (B) Multiply both by $100$: $14.4\div12=1.2$.
14. $25$ cm expressed in metres is:
  1. $2.5$ m
  2. $0.25$ m
  3. $0.025$ m
  4. $250$ m
Answer: (B) $1$ cm $=0.01$ m, so $25$ cm $=0.25$ m.
15. $3.4\times2.5$ equals:
  1. $8.5$
  2. $85$
  3. $0.85$
  4. $6.5$
Answer: (A) $34\times25=850$, with $1+1=2$ places: $8.50=8.5$.
Important Questions
Q1. Write $0.625$ as a fraction in lowest terms. (2 marks)
Answer: $0.625=\dfrac{625}{1000}$. Divide top and bottom by their HCF $125$: $\dfrac{625}{1000}=\dfrac{5}{8}$.
Q2. Arrange in ascending order: $0.8,\ 0.08,\ 0.808,\ 0.88$. (2 marks)
Answer: Write with equal places: $0.800,\ 0.080,\ 0.808,\ 0.880$. Ascending: $0.08<0.8<0.808<0.88$.
Q3. A rope is $15.6$ m long. It is cut into pieces of $1.3$ m each. How many pieces are obtained? (3 marks)
Answer: Number of pieces $=15.6\div1.3$. Multiply both by $10$: $156\div13=12$. So $12$ pieces are obtained.
Q4. Reshma buys $3.5$ kg of rice at ₹$48.40$ per kg. Find the total cost. (3 marks)
Answer: Cost $=48.40\times3.5$. Multiply as whole numbers: $4840\times35=169400$. Total decimal places $=2+1=3$, so $169.400=$ ₹$169.40$.
Q5. Explain, with an example, how multiplying and dividing a decimal by $100$ moves the decimal point. (3 marks)
Answer: Multiplying by $100$ moves the decimal point two places to the right: $3.45\times100=345$. Dividing by $100$ moves the point two places to the left: $3.45\div100=0.0345$. The number of zeros equals the number of places the point shifts.
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More Class 7 Mathematics chapters
Large Numbers Around Us · Arithmetic Expressions · A Peek Beyond the Point · Expressions using Letter-Numbers · Parallel and Intersecting Lines · Number Play · A Tale of Three Intersecting Lines · Working with Fractions
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