Important Formulas & Points to Remember: A Peek Beyond the Point | Mathematics (Ganita Prakash) Class 7 - New NCERT PDF Download

 Page 1


1 T h e N e e d f o r S m a l l e r U n i t s
• Wh y Smaller Units? When precise measurements are needed, units are divided into smaller
parts (e.g., centimeters into millimeters) to measure small differences accur ately .
• Reading Measurements :
– 2
7
10
cm is read as “two and seven-tenth centimeters.”
– 3
2
10
cm is read as “three and two-tenth centimeters.”
• Measurement Process : Use a ruler with smaller divisions (e.g., millimeters) to measure
objects lik e screws accur ately .
2 A T e n t h P a r t
• Fr actional Units :
– A unit divided into 10 equal parts gives “one-tenths” (
1
10
).
– Example: 3
4
10
units = 3+
4
10
=
34
10
units.
• A ddition of Fr actional Units :
– Example: 2
7
10
+3
6
10
= (2+3)+
(
7
10
+
6
10
)
= 5+
13
10
= 6
3
10
units.
– Alternative: Convert to tenths:
27
10
+
36
10
=
62
10
= 6
2
10
units.
• Subtr action of Fr actional Units :
– Example: 12
4
10
-6
7
10
= (12-6)+
(
4
10
-
7
10
)
= 6-
3
10
= 5
7
10
units.
– Alternative: Convert to tenths:
124
10
-
67
10
=
57
10
= 5
7
10
units.
• Ordering Lengths : Convert to a common form (e.g., tenths) to arr ange lengths lik e
9
10
,1
7
10
,
13
10
, etc., in increasing order .
3 A H u n d r e d t h P a r t
• Hundredths : Each one-tenth divided into 10 equal parts gives “one-hundredths” (
1
100
).
– Example: 4
4
10
5
100
= 4+
4
10
+
5
100
= 4
45
100
units.
• A ddition with Hundredths :
– Example: 15
3
10
4
100
+2
6
10
8
100
= (15+2)+
(
3
10
+
6
10
)
+
(
4
100
+
8
100
)
= 17+
9
10
+
12
100
= 18
2
100
units.
– Alternative: Convert to hundredths:
1534
100
+
268
100
=
1802
100
= 18
2
100
units.
• Subtr action with Hundredths :
– Example: 15
3
10
4
100
-2
6
10
8
100
= (15-2)+
(
3
10
-
6
10
)
+
(
4
100
-
8
100
)
= 13-
3
10
-
4
100
= 12
6
10
6
100
units.
– Alternative: Convert to hundredths:
1534
100
-
268
100
=
1266
100
= 12
66
100
= 12
6
10
6
100
units.
1
Page 2


1 T h e N e e d f o r S m a l l e r U n i t s
• Wh y Smaller Units? When precise measurements are needed, units are divided into smaller
parts (e.g., centimeters into millimeters) to measure small differences accur ately .
• Reading Measurements :
– 2
7
10
cm is read as “two and seven-tenth centimeters.”
– 3
2
10
cm is read as “three and two-tenth centimeters.”
• Measurement Process : Use a ruler with smaller divisions (e.g., millimeters) to measure
objects lik e screws accur ately .
2 A T e n t h P a r t
• Fr actional Units :
– A unit divided into 10 equal parts gives “one-tenths” (
1
10
).
– Example: 3
4
10
units = 3+
4
10
=
34
10
units.
• A ddition of Fr actional Units :
– Example: 2
7
10
+3
6
10
= (2+3)+
(
7
10
+
6
10
)
= 5+
13
10
= 6
3
10
units.
– Alternative: Convert to tenths:
27
10
+
36
10
=
62
10
= 6
2
10
units.
• Subtr action of Fr actional Units :
– Example: 12
4
10
-6
7
10
= (12-6)+
(
4
10
-
7
10
)
= 6-
3
10
= 5
7
10
units.
– Alternative: Convert to tenths:
124
10
-
67
10
=
57
10
= 5
7
10
units.
• Ordering Lengths : Convert to a common form (e.g., tenths) to arr ange lengths lik e
9
10
,1
7
10
,
13
10
, etc., in increasing order .
3 A H u n d r e d t h P a r t
• Hundredths : Each one-tenth divided into 10 equal parts gives “one-hundredths” (
1
100
).
– Example: 4
4
10
5
100
= 4+
4
10
+
5
100
= 4
45
100
units.
• A ddition with Hundredths :
– Example: 15
3
10
4
100
+2
6
10
8
100
= (15+2)+
(
3
10
+
6
10
)
+
(
4
100
+
8
100
)
= 17+
9
10
+
12
100
= 18
2
100
units.
– Alternative: Convert to hundredths:
1534
100
+
268
100
=
1802
100
= 18
2
100
units.
• Subtr action with Hundredths :
– Example: 15
3
10
4
100
-2
6
10
8
100
= (15-2)+
(
3
10
-
6
10
)
+
(
4
100
-
8
100
)
= 13-
3
10
-
4
100
= 12
6
10
6
100
units.
– Alternative: Convert to hundredths:
1534
100
-
268
100
=
1266
100
= 12
66
100
= 12
6
10
6
100
units.
1
4 D e c i m a l P l a c e V a l u e
• Decimal S ystem : Based on powers of 10, extending the Indian place value system to fr ac-
tional parts.
– Example: 705 = 7×100+5×1 ; 70.5 = 7×10+5×
1
10
; 7.05 = 7×1+5×
1
100
.
• Reading Decimals :
– 70.5 is read as “seventy point five” ( 70+
5
10
).
– 7.05 is read as “seven point zero five” ( 7+
5
100
).
– 0.274 is read as “zero point two seven four ” (
2
10
+
7
100
+
4
1000
).
• Writing Decimals :
– Example: 234 tenths =
234
10
= 23.4 .
– Example: 234 hundredths =
234
100
= 2.34 .
5 U n i t s o f M e a s u r e m e n t
• Length Conversions :
– 1 cm = 10 mm; 1 mm = 0.1 cm.
– 1 m = 100 cm; 1 cm = 0.01 m.
– 1 m = 1000 mm; 1 mm = 0.001 m.
– Examples:
*
5 mm = 0.5 cm.
*
12 mm = 1.2 cm.
*
5.6 cm = 56 mm.
*
10 cm = 0.1 m.
*
15 cm = 0.15 m.
• W eight Conversions :
– 1 k g = 1000 g; 1 g = 0.001 k g.
– 1 g = 1000 mg; 1 mg = 0.001 g.
– Examples:
*
5 g = 0.005 k g.
*
254 g = 0.254 k g.
• Rupee-Paise Conversions :
– 1 rupee = 100 paise; 1 paisa = 0.01 rupee.
– Examples:
*
75 paise = 0.75 rupee.
*
10 paise = 0.10 rupee.
6 L o c a t i n g a n d C o m p a r i n g D e c i m a l s
• Locating Decimals on a Number Line :
– A decimal lik e 1.4 = 1 +
4
10
lies between 1 and 2, where the unit between 1 and 2 is
divided into 10 equal parts, and 4 parts are tak en.
– Example: T o locate1.4 , divide the interval between 1 and 2 into 10 equal parts, marking
1.1,1.2,...,1.9 , and select 1.4 .
– F or finer precision (e.g., 4.185), magnify segments (e.g., 4 to 5, then 4.1 to 4.2, then 4.18
to 4.19) to pinpoint the exact position.
2
Page 3


1 T h e N e e d f o r S m a l l e r U n i t s
• Wh y Smaller Units? When precise measurements are needed, units are divided into smaller
parts (e.g., centimeters into millimeters) to measure small differences accur ately .
• Reading Measurements :
– 2
7
10
cm is read as “two and seven-tenth centimeters.”
– 3
2
10
cm is read as “three and two-tenth centimeters.”
• Measurement Process : Use a ruler with smaller divisions (e.g., millimeters) to measure
objects lik e screws accur ately .
2 A T e n t h P a r t
• Fr actional Units :
– A unit divided into 10 equal parts gives “one-tenths” (
1
10
).
– Example: 3
4
10
units = 3+
4
10
=
34
10
units.
• A ddition of Fr actional Units :
– Example: 2
7
10
+3
6
10
= (2+3)+
(
7
10
+
6
10
)
= 5+
13
10
= 6
3
10
units.
– Alternative: Convert to tenths:
27
10
+
36
10
=
62
10
= 6
2
10
units.
• Subtr action of Fr actional Units :
– Example: 12
4
10
-6
7
10
= (12-6)+
(
4
10
-
7
10
)
= 6-
3
10
= 5
7
10
units.
– Alternative: Convert to tenths:
124
10
-
67
10
=
57
10
= 5
7
10
units.
• Ordering Lengths : Convert to a common form (e.g., tenths) to arr ange lengths lik e
9
10
,1
7
10
,
13
10
, etc., in increasing order .
3 A H u n d r e d t h P a r t
• Hundredths : Each one-tenth divided into 10 equal parts gives “one-hundredths” (
1
100
).
– Example: 4
4
10
5
100
= 4+
4
10
+
5
100
= 4
45
100
units.
• A ddition with Hundredths :
– Example: 15
3
10
4
100
+2
6
10
8
100
= (15+2)+
(
3
10
+
6
10
)
+
(
4
100
+
8
100
)
= 17+
9
10
+
12
100
= 18
2
100
units.
– Alternative: Convert to hundredths:
1534
100
+
268
100
=
1802
100
= 18
2
100
units.
• Subtr action with Hundredths :
– Example: 15
3
10
4
100
-2
6
10
8
100
= (15-2)+
(
3
10
-
6
10
)
+
(
4
100
-
8
100
)
= 13-
3
10
-
4
100
= 12
6
10
6
100
units.
– Alternative: Convert to hundredths:
1534
100
-
268
100
=
1266
100
= 12
66
100
= 12
6
10
6
100
units.
1
4 D e c i m a l P l a c e V a l u e
• Decimal S ystem : Based on powers of 10, extending the Indian place value system to fr ac-
tional parts.
– Example: 705 = 7×100+5×1 ; 70.5 = 7×10+5×
1
10
; 7.05 = 7×1+5×
1
100
.
• Reading Decimals :
– 70.5 is read as “seventy point five” ( 70+
5
10
).
– 7.05 is read as “seven point zero five” ( 7+
5
100
).
– 0.274 is read as “zero point two seven four ” (
2
10
+
7
100
+
4
1000
).
• Writing Decimals :
– Example: 234 tenths =
234
10
= 23.4 .
– Example: 234 hundredths =
234
100
= 2.34 .
5 U n i t s o f M e a s u r e m e n t
• Length Conversions :
– 1 cm = 10 mm; 1 mm = 0.1 cm.
– 1 m = 100 cm; 1 cm = 0.01 m.
– 1 m = 1000 mm; 1 mm = 0.001 m.
– Examples:
*
5 mm = 0.5 cm.
*
12 mm = 1.2 cm.
*
5.6 cm = 56 mm.
*
10 cm = 0.1 m.
*
15 cm = 0.15 m.
• W eight Conversions :
– 1 k g = 1000 g; 1 g = 0.001 k g.
– 1 g = 1000 mg; 1 mg = 0.001 g.
– Examples:
*
5 g = 0.005 k g.
*
254 g = 0.254 k g.
• Rupee-Paise Conversions :
– 1 rupee = 100 paise; 1 paisa = 0.01 rupee.
– Examples:
*
75 paise = 0.75 rupee.
*
10 paise = 0.10 rupee.
6 L o c a t i n g a n d C o m p a r i n g D e c i m a l s
• Locating Decimals on a Number Line :
– A decimal lik e 1.4 = 1 +
4
10
lies between 1 and 2, where the unit between 1 and 2 is
divided into 10 equal parts, and 4 parts are tak en.
– Example: T o locate1.4 , divide the interval between 1 and 2 into 10 equal parts, marking
1.1,1.2,...,1.9 , and select 1.4 .
– F or finer precision (e.g., 4.185), magnify segments (e.g., 4 to 5, then 4.1 to 4.2, then 4.18
to 4.19) to pinpoint the exact position.
2
• Zero Dilemma :
– A dding zeros to the right of a decimal does not change its value: 0.2 = 0.20 = 0.200 (all
represent
2
10
).
– However , 0.2?= 0.02?= 0.002, as they represent
2
10
,
2
100
, and
2
1000
, respectively .
• Comparing Decimals :
– Compare digits step-b y-step b y place value, starting with the highest (units, then tenths,
hundredths, etc.).
– Example: F or 6.456 vs. 6.465:
*
Units: 6 = 6 .
*
T enths: 4 = 4 .
*
Hundredths: 5 < 6 , so 6.456 < 6.465.
– Example: Compare 1.23 vs. 1.32 :
*
Units: 1 = 1 .
*
T enths: 2 < 3 , so 1.23 < 1.32 .
• Closest Decimals :
– T o find the decimal closest to a number , compute the absolute difference.
– Example: F or 0.9,1.1,1.01,1.11 relative to 1:
*
|1-0.9| = 0.1 ,|1-1.01| = 0.01 ,|1-1.1| = 0.1 ,|1-1.11| = 0.11 .
*
1.01 is closest to 1 (difference of 0.01 ).
7 A d d i t i o n a n d S u b t r a c t i o n o f D e c i m a l s
• A ddition :
– Align decimals b y place value and add as with whole numbers.
– Example: 2.7+3.5 = 6.2 , or in fr actions: 2
7
10
+3
5
10
= 5
12
10
= 6
2
10
.
– Detailed place value for 75.345+86.691:
(7×10+5×1+3×
1
10
+4×
1
100
+5×
1
1000
)+(8×10+6×1+6×
1
10
+9×
1
100
+1×
1
1000
) = 162.036
• Subtr action :
– Align decimals b y place value and subtr act as with whole numbers.
– Example: 3.5-2.7 = 0.8 , or in fr actions: 3
5
10
-2
7
10
=
8
10
= 0.8 .
– Detailed place value for 84.691-77.345:
(8×10+4×1+6×
1
10
+9×
1
100
+1×
1
1000
)-(7×10+7×1+3×
1
10
+4×
1
100
+5×
1
1000
) = 7.346
• Estimating Sums and Differences :
– The sum of two decimals is greater than the sum of their whole number parts and less
than the sum of their whole number parts plus 2.
– Example: F or 25.936+8.202, the sum is > 25+8 = 33 and < 25+1+8+1 = 35 . A ctual
sum: 34.138 .
– F or differences, the result lies between the difference of whole number parts and that
difference plus or minus 1.
3
Page 4


1 T h e N e e d f o r S m a l l e r U n i t s
• Wh y Smaller Units? When precise measurements are needed, units are divided into smaller
parts (e.g., centimeters into millimeters) to measure small differences accur ately .
• Reading Measurements :
– 2
7
10
cm is read as “two and seven-tenth centimeters.”
– 3
2
10
cm is read as “three and two-tenth centimeters.”
• Measurement Process : Use a ruler with smaller divisions (e.g., millimeters) to measure
objects lik e screws accur ately .
2 A T e n t h P a r t
• Fr actional Units :
– A unit divided into 10 equal parts gives “one-tenths” (
1
10
).
– Example: 3
4
10
units = 3+
4
10
=
34
10
units.
• A ddition of Fr actional Units :
– Example: 2
7
10
+3
6
10
= (2+3)+
(
7
10
+
6
10
)
= 5+
13
10
= 6
3
10
units.
– Alternative: Convert to tenths:
27
10
+
36
10
=
62
10
= 6
2
10
units.
• Subtr action of Fr actional Units :
– Example: 12
4
10
-6
7
10
= (12-6)+
(
4
10
-
7
10
)
= 6-
3
10
= 5
7
10
units.
– Alternative: Convert to tenths:
124
10
-
67
10
=
57
10
= 5
7
10
units.
• Ordering Lengths : Convert to a common form (e.g., tenths) to arr ange lengths lik e
9
10
,1
7
10
,
13
10
, etc., in increasing order .
3 A H u n d r e d t h P a r t
• Hundredths : Each one-tenth divided into 10 equal parts gives “one-hundredths” (
1
100
).
– Example: 4
4
10
5
100
= 4+
4
10
+
5
100
= 4
45
100
units.
• A ddition with Hundredths :
– Example: 15
3
10
4
100
+2
6
10
8
100
= (15+2)+
(
3
10
+
6
10
)
+
(
4
100
+
8
100
)
= 17+
9
10
+
12
100
= 18
2
100
units.
– Alternative: Convert to hundredths:
1534
100
+
268
100
=
1802
100
= 18
2
100
units.
• Subtr action with Hundredths :
– Example: 15
3
10
4
100
-2
6
10
8
100
= (15-2)+
(
3
10
-
6
10
)
+
(
4
100
-
8
100
)
= 13-
3
10
-
4
100
= 12
6
10
6
100
units.
– Alternative: Convert to hundredths:
1534
100
-
268
100
=
1266
100
= 12
66
100
= 12
6
10
6
100
units.
1
4 D e c i m a l P l a c e V a l u e
• Decimal S ystem : Based on powers of 10, extending the Indian place value system to fr ac-
tional parts.
– Example: 705 = 7×100+5×1 ; 70.5 = 7×10+5×
1
10
; 7.05 = 7×1+5×
1
100
.
• Reading Decimals :
– 70.5 is read as “seventy point five” ( 70+
5
10
).
– 7.05 is read as “seven point zero five” ( 7+
5
100
).
– 0.274 is read as “zero point two seven four ” (
2
10
+
7
100
+
4
1000
).
• Writing Decimals :
– Example: 234 tenths =
234
10
= 23.4 .
– Example: 234 hundredths =
234
100
= 2.34 .
5 U n i t s o f M e a s u r e m e n t
• Length Conversions :
– 1 cm = 10 mm; 1 mm = 0.1 cm.
– 1 m = 100 cm; 1 cm = 0.01 m.
– 1 m = 1000 mm; 1 mm = 0.001 m.
– Examples:
*
5 mm = 0.5 cm.
*
12 mm = 1.2 cm.
*
5.6 cm = 56 mm.
*
10 cm = 0.1 m.
*
15 cm = 0.15 m.
• W eight Conversions :
– 1 k g = 1000 g; 1 g = 0.001 k g.
– 1 g = 1000 mg; 1 mg = 0.001 g.
– Examples:
*
5 g = 0.005 k g.
*
254 g = 0.254 k g.
• Rupee-Paise Conversions :
– 1 rupee = 100 paise; 1 paisa = 0.01 rupee.
– Examples:
*
75 paise = 0.75 rupee.
*
10 paise = 0.10 rupee.
6 L o c a t i n g a n d C o m p a r i n g D e c i m a l s
• Locating Decimals on a Number Line :
– A decimal lik e 1.4 = 1 +
4
10
lies between 1 and 2, where the unit between 1 and 2 is
divided into 10 equal parts, and 4 parts are tak en.
– Example: T o locate1.4 , divide the interval between 1 and 2 into 10 equal parts, marking
1.1,1.2,...,1.9 , and select 1.4 .
– F or finer precision (e.g., 4.185), magnify segments (e.g., 4 to 5, then 4.1 to 4.2, then 4.18
to 4.19) to pinpoint the exact position.
2
• Zero Dilemma :
– A dding zeros to the right of a decimal does not change its value: 0.2 = 0.20 = 0.200 (all
represent
2
10
).
– However , 0.2?= 0.02?= 0.002, as they represent
2
10
,
2
100
, and
2
1000
, respectively .
• Comparing Decimals :
– Compare digits step-b y-step b y place value, starting with the highest (units, then tenths,
hundredths, etc.).
– Example: F or 6.456 vs. 6.465:
*
Units: 6 = 6 .
*
T enths: 4 = 4 .
*
Hundredths: 5 < 6 , so 6.456 < 6.465.
– Example: Compare 1.23 vs. 1.32 :
*
Units: 1 = 1 .
*
T enths: 2 < 3 , so 1.23 < 1.32 .
• Closest Decimals :
– T o find the decimal closest to a number , compute the absolute difference.
– Example: F or 0.9,1.1,1.01,1.11 relative to 1:
*
|1-0.9| = 0.1 ,|1-1.01| = 0.01 ,|1-1.1| = 0.1 ,|1-1.11| = 0.11 .
*
1.01 is closest to 1 (difference of 0.01 ).
7 A d d i t i o n a n d S u b t r a c t i o n o f D e c i m a l s
• A ddition :
– Align decimals b y place value and add as with whole numbers.
– Example: 2.7+3.5 = 6.2 , or in fr actions: 2
7
10
+3
5
10
= 5
12
10
= 6
2
10
.
– Detailed place value for 75.345+86.691:
(7×10+5×1+3×
1
10
+4×
1
100
+5×
1
1000
)+(8×10+6×1+6×
1
10
+9×
1
100
+1×
1
1000
) = 162.036
• Subtr action :
– Align decimals b y place value and subtr act as with whole numbers.
– Example: 3.5-2.7 = 0.8 , or in fr actions: 3
5
10
-2
7
10
=
8
10
= 0.8 .
– Detailed place value for 84.691-77.345:
(8×10+4×1+6×
1
10
+9×
1
100
+1×
1
1000
)-(7×10+7×1+3×
1
10
+4×
1
100
+5×
1
1000
) = 7.346
• Estimating Sums and Differences :
– The sum of two decimals is greater than the sum of their whole number parts and less
than the sum of their whole number parts plus 2.
– Example: F or 25.936+8.202, the sum is > 25+8 = 33 and < 25+1+8+1 = 35 . A ctual
sum: 34.138 .
– F or differences, the result lies between the difference of whole number parts and that
difference plus or minus 1.
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8 M o r e o n t h e D e c i m a l S y s t e m
• Decimal and Measurement Disasters :
– Errors in decimal point placement or unit conversions can lead to significant issues
(e.g., Amsterdam’ s €188 million error due to processing in euro cents instead of euros;
Air Canada’ s 1983 fuel miscalculation).
– Medical errors: Misreading 0.05 mg as 0.5 mg can res ult in a 10-fold overdose.
• Deceptive Decimal Notation :
– Example: 4.5 hours = 4 hours + 0.5×60 = 30 minutes, so 4 : 30 p.m., not 4 : 05 or 4 : 50 .
– In crick et, 5.5 overs = 5 overs + 5 balls (since 1 over = 6 balls).
• Conversions :
– Fr actions to decimals:
5
100
= 0.05 ,
16
1000
= 0.016 ,
12
10
= 1.2 ,
254
1000
= 0.254.
– Decimals to fr actions: 0.34 =
3
10
+
4
100
, 0.362 =
3
10
+
6
100
+
2
1000
.
– Example: 1 km = 1,000,000 mm.
– Example: Insur ance fee for 100,000 passengers at 45 paise each: 100,000 × 0.45 =
45,000 rupees.
K ey Poi nts to Remember
• Precision in Measurement : Smaller units (tenths, hundredths) allow for more accur ate
measurements.
• Decimal Notation : Uses a decimal point to separ ate whole numbers from fr actional parts,
extending the place value system.
• Conversions : Understanding conversions between units (mm, cm, m, g, k g, paise, rupees)
is crucial for pr actical applications.
• Locating Decimals : Use number lines with magnified segments for precise placement of
decimals lik e 4.185 or 9.876.
• Comparing Decimals : Compare digits b y place value, stopping at the first differing digit to
determine which number is larger .
• Arithmetic with Decimals : A dd and subtr act b y aligning place values, similar to whole
numbers; verify with fr actional forms (e.g., tenths).
• Decimal Equivalence : Tr ailing zeros after the decimal point do not change the value (e.g.,
0.2 = 0.20 ).
• Precision and Units : Correct decimal point placement and unit conversions are critical to
avoid errors in real-world applications.
• Decimal Sequences : Identify patterns (e.g., adding0.4 in4.4,4.8,5.2,... ) to extend sequences.
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