A Peek Beyond the Point — Class 7 Mathematics Notes (NCERT)

Snapshot

This chapter goes "beyond the point" — into decimals as a way of writing tenths, hundredths and thousandths.
Each place after the decimal point is ten times smaller than the one before: tenths $\tfrac{1}{10}$, hundredths $\tfrac{1}{100}$, thousandths $\tfrac{1}{1000}$.
Decimals and fractions are two views of the same number: $0.7=\tfrac{7}{10}$, $0.25=\tfrac{25}{100}=\tfrac14$.
You learn to compare, order, add, subtract, multiply and divide decimals, and to round them.
Place value extends smoothly: $34.56=3\times10+4+5\times\tfrac{1}{10}+6\times\tfrac{1}{100}$.
Decimals appear everywhere — money (₹), length (m, cm), mass (kg, g) — making unit conversion easy.
Weightage: ~9 marks/year — operations on decimals, comparisons, and measurement word problems.

Detailed Notes

1. Why look beyond the point?

Whole numbers are not enough for real life: prices like ₹$45.75$, a height of $1.62$ m, or a race time of $10.85$ seconds all need numbers between the whole numbers. A decimal uses a small dot — the decimal point — to separate the whole part from the fractional part. The digits after the point measure parts of one, in tenths, hundredths and so on. This chapter explores how these decimal places work and how to calculate with them.

2. Decimal place value

Just as every place to the left is ten times bigger, every place to the right of the point is ten times smaller.

Tens	Ones	.	Tenths	Hundredths	Thousandths
3	4	.	5	6	8

$34.568 = 3\times10 + 4\times1 + 5\times\dfrac{1}{10} + 6\times\dfrac{1}{100} + 8\times\dfrac{1}{1000}$

So the digit $5$ here is worth $\tfrac{5}{10}=0.5$, the $6$ is worth $\tfrac{6}{100}=0.06$, and the $8$ is worth $\tfrac{8}{1000}=0.008$.

3. Decimals and fractions — two views

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

$0.3=\dfrac{3}{10}$
$0.47=\dfrac{47}{100}$
$0.125=\dfrac{125}{1000}=\dfrac{1}{8}$

Worked example — fraction to decimal

$\dfrac{3}{4}=\dfrac{3\times25}{4\times25}=\dfrac{75}{100}=0.75.$ Make the denominator $100$, then read off.

Worked example — decimal to fraction (lowest terms)

$0.6=\dfrac{6}{10}=\dfrac{3}{5}.$ Always reduce to lowest terms.

4. Comparing and ordering decimals

To compare decimals, compare digit by digit from the left, place by place. It helps to make the number of decimal places equal by adding trailing zeros (which do not change the value).

Worked example — which is greater, $0.7$ or $0.65$?

Write $0.7=0.70$. Compare tenths: $7>6$, so $0.7>0.65.$ (A common error is to think $0.65$ is bigger because it has more digits.)

Worked example — order ascending

$0.5,\ 0.45,\ 0.405,\ 0.54.$ Pad to three places: $0.500,0.450,0.405,0.540.$ Ascending: $0.405 < 0.45 < 0.5 < 0.54.$

5. Adding and subtracting decimals

Golden rule: line up the decimal points (so tenths sit under tenths, etc.), pad with zeros, then add or subtract as with whole numbers, keeping the point in the same column.

Worked example — add $12.4+3.75$

Write $12.40 + 3.75$. Add: $12.40+3.75=16.15.$

Worked example — subtract $20-6.85$

Write $20.00 - 6.85 = 13.15.$ The whole number is padded to two decimal places.

Always align the decimal points before adding or subtracting.

6. Multiplying decimals

Multiply ignoring the points, then place the point so the answer has as many decimal places as the total in the two factors.

Worked example — $2.5\times0.4$

$25\times4=100$. Total decimal places $=1+1=2$, so answer $=1.00=1.$

Worked example — multiply by $10,100,1000$

Move the point right by the number of zeros: $3.456\times10=34.56$; $3.456\times100=345.6$; $3.456\times1000=3456.$

Decimal places in product $=$ sum of decimal places in the factors.

7. Dividing decimals

Dividing by $10,100,1000$: move the point left by the number of zeros.

Worked example

$56.4\div10=5.64$; $56.4\div100=0.564$; $56.4\div1000=0.0564.$

Dividing a decimal by a whole number: divide as usual, bringing the point straight up into the quotient.

Worked example — $9.6\div4$

$9.6\div4 = 2.4$ (since $4\times2.4=9.6$).

Dividing by a decimal: shift the point in both numbers so the divisor becomes a whole number.

Worked example — $4.5\div0.5$

Multiply both by $10$: $45\div5=9.$

8. Rounding decimals

To round to a given decimal place, look at the next digit: $5$ or more → round up, else keep.

Worked example — round to one decimal place

$7.86 \to$ next digit $6\ (\ge5)$, so $7.9.$ $4.32 \to$ next digit $2\ (<5)$, so $4.3.$

Worked example — round ₹$45.728$ to the nearest paisa (two places)

Next digit is $8\ (\ge5)$, so ₹$45.73.$

9. Decimals in measurement and money

Decimals make unit conversion within the metric system simple, because units go in powers of $10$.

Conversion	Example
$1$ m $=100$ cm	$2.5$ m $=250$ cm
$1$ cm $=\tfrac{1}{100}$ m	$75$ cm $=0.75$ m
$1$ kg $=1000$ g	$1.2$ kg $=1200$ g
$1$ rupee $=100$ paise	₹$3.50=350$ paise

Worked example — total bill

Items cost ₹$23.50$, ₹$8.75$ and ₹$12.25$. Total $=23.50+8.75+12.25=44.50.$ So the bill is ₹$44.50$.

Worked example — length problem

A ribbon $3.6$ m long is cut into $4$ equal pieces. Each piece $=3.6\div4=0.9$ m $=90$ cm.

10. Like and unlike decimals; the number line

Decimals with the same number of decimal places are called like decimals ($0.45$ and $0.78$); those with different numbers are unlike ($0.4$ and $0.456$). We can always turn unlike decimals into like decimals by writing extra zeros at the end, because trailing zeros do not change a decimal's value: $0.4=0.40=0.400$. This is the single most useful habit for comparing, adding and subtracting safely.

Every decimal also has a home on the number line. Between $0$ and $1$ we can mark tenths $0.1,0.2,\dots,0.9$. To place $0.65$, we zoom into the gap between $0.6$ and $0.7$ and split it into ten equal parts; $0.65$ sits exactly halfway. Zooming further lets us place thousandths. This picture explains why $0.7$ lies to the right of $0.65$ — it is the right end of the same little interval, hence larger.

Worked example — place $1.25$ on the number line

$1.25$ lies between $1$ and $2$, and within that, between $1.2$ and $1.3$, exactly at the midpoint. So it is a quarter of the way from $1$ to $2$, matching $1\tfrac14$.

11. Patterns and estimation with decimals

Estimating decimal answers protects you from silly errors. Round each decimal to the nearest whole number, do the easy sum, and check your exact answer is close.

Worked example — estimate then compute

Estimate $9.8+4.1$ as $10+4=14$; the exact value $13.9$ is close, so it is reasonable.

Watch the lovely pattern when you keep dividing by $10$: $5.6,\ 0.56,\ 0.056,\ 0.0056,\dots$ — the digits stay the same and only the point marches left. The mirror pattern appears when you multiply by $10$ repeatedly. Recognising such patterns makes mental arithmetic with decimals fast and reliable.

12. Common mistakes to avoid

Thinking more digits after the point means a bigger number ($0.65 < 0.7$).
Not aligning the decimal points when adding/subtracting.
Forgetting to count total decimal places when multiplying.
Moving the point the wrong way: $\times10$ moves it right, $\div10$ moves it left.
Dropping the decimal point in the quotient during division.

13. Quick revision checklist

Places after point: tenths, hundredths, thousandths ($\tfrac{1}{10},\tfrac{1}{100},\tfrac{1}{1000}$).
Decimal $=$ fraction with denominator $10,100,1000,\dots$
Compare left to right; pad with trailing zeros.
Add/subtract: line up the points.
Multiply: add the decimal places; $\times10^{n}$ moves point right, $\div10^{n}$ moves it left.

Practice MCQs

1. The place value of $7$ in $5.07$ is:

$\dfrac{7}{10}$
$\dfrac{7}{100}$
$7$
$\dfrac{7}{1000}$

Answer: (B) $7$ is in the hundredths place, so its value is $\tfrac{7}{100}=0.07$.

2. $\dfrac{3}{4}$ as a decimal is:

$0.34$
$0.75$
$0.43$
$0.7$

Answer: (B) $\tfrac{3}{4}=\tfrac{75}{100}=0.75$.

3. Which is the greatest?

$0.6$
$0.59$
$0.506$
$0.56$

Answer: (A) Compare tenths: $0.6$ has $6$ tenths, the largest.

4. $4.5\times0.2$ equals:

$0.9$
$9$
$0.09$
$90$

Answer: (A) $45\times2=90$; two decimal places → $0.90=0.9$.

5. $12.5\div10$ equals:

$125$
$1.25$
$0.125$
$12.5$

Answer: (B) Dividing by $10$ moves the point one place left.

6. $0.25$ as a fraction in lowest terms is:

$\dfrac{25}{10}$
$\dfrac{1}{4}$
$\dfrac{1}{5}$
$\dfrac{2}{5}$

Answer: (B) $0.25=\tfrac{25}{100}=\tfrac14$.

7. $3.456\times100$ equals:

$34.56$
$345.6$
$3456$
$0.3456$

Answer: (B) Multiplying by $100$ moves the point two places right.

8. $7.2 + 0.85$ equals:

$7.92$
$8.05$
$0.157$
$8.5$

Answer: (B) $7.20+0.85=8.05$ (align the points).

9. $75$ cm expressed in metres is:

$7.5$ m
$0.75$ m
$0.075$ m
$750$ m

Answer: (B) $1$ cm $=0.01$ m, so $75$ cm $=0.75$ m.

10. $4.5\div0.5$ equals:

$0.9$
$9$
$90$
$0.09$

Answer: (B) Multiply both by $10$: $45\div5=9$.

11. $6.78$ rounded to one decimal place is:

$6.7$
$6.8$
$6.0$
$7.0$

Answer: (B) Next digit $8\ (\ge5)$, so round up to $6.8$.

12. Which of these equals $0.5$?

$\dfrac{1}{5}$
$\dfrac{5}{100}$
$\dfrac{1}{2}$
$\dfrac{2}{5}$

Answer: (C) $\tfrac12=\tfrac{5}{10}=0.5$.

13. The expanded form of $2.34$ is:

$2+\dfrac{3}{10}+\dfrac{4}{100}$
$2+\dfrac{3}{100}+\dfrac{4}{10}$
$2+3+4$
$2+\dfrac{34}{10}$

Answer: (A) $3$ is tenths, $4$ is hundredths.

14. ₹$3.50$ in paise is:

$35$ paise
$350$ paise
$3.5$ paise
$3500$ paise

Answer: (B) $1$ rupee $=100$ paise, so ₹$3.50=350$ paise.

15. $1.2$ kg in grams is:

$12$ g
$120$ g
$1200$ g
$12000$ g

Answer: (C) $1$ kg $=1000$ g, so $1.2$ kg $=1200$ g.

Important Questions

Q1. Arrange in descending order: $0.7,\ 0.07,\ 0.77,\ 0.707.$ (2 marks)

Answer: Pad to three places: $0.700,0.070,0.770,0.707.$ Descending: $0.77 > 0.707 > 0.7 > 0.07.$

Q2. A rope is $15.5$ m long. Pieces of $2.75$ m are cut off. After cutting two pieces, how much remains? (3 marks)

Answer: Two pieces $=2.75\times2=5.50$ m. Remaining $=15.50-5.50=10.0$ m.

Q3. Convert $\dfrac{7}{8}$ into a decimal. (2 marks)

Answer: $\tfrac{7}{8}=\tfrac{7\times125}{1000}=\tfrac{875}{1000}=0.875.$ (Or divide $7\div8=0.875$.)

Q4. Find the value of $3.6\times2.5$ and explain the placement of the decimal point. (2 marks)

Answer: $36\times25=900$. Total decimal places $=1+1=2$, so $3.6\times2.5=9.00=9$.

Q5. The total mass of $5$ identical books is $4.25$ kg. Find the mass of one book in kg and g. (3 marks)

Answer: One book $=4.25\div5=0.85$ kg $=850$ g.

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More Class 7 Mathematics chapters

Large Numbers Around Us · Arithmetic Expressions · Expressions using Letter-Numbers · Parallel and Intersecting Lines · Number Play · A Tale of Three Intersecting Lines · Working with Fractions · Geometric Twins