Page 1
Q u e s t i o n : 1
Express each of the following ratios in simplest form:
i
24 : 40
ii
13.5 : 15
iii
6
2
3
: 7
1
2
iv
1
6
:
1
9
v
4: 5:
9
2
vi
2.5 : 6.5 : 8
S o l u t i o n :
i
HCF of 24 and 40 is 8.
? 24 : 40 =
24
40
=
24 ÷ 8
40 ÷ 8
=
3
5
= 3 : 5
Hence, 24 : 40 in its simplest form is 3 : 5.
ii
HCF of 13.5 and 15 is 1.5.
13.5
15
=
135
150
The HCF of 135 and 150 is 15. =
135 ÷ 15
150 ÷ 15
=
9
10
Hence, 13.5 : 15 in its simplest form is 9 : 10.
iii
20
3
:
15
2
= 40 : 45
The HCF of 40 and 45 is 5.
? 40 : 45 =
40
45
=
40 ÷ 5
45 ÷ 5
=
8
9
= 8 : 9
Hence, 6
2
3
: 7
1
2
in its simplest form is 8 : 9
(iv) 9 : 6
The HCF of 9 and 6 is 3.
? 9 : 6 =
9
6
=
9 ÷ 3
6 ÷ 3
= 3 : 2
Hence,
1
6
:
1
9
in its simplest form is 3 : 2.
(v) LCM of the denominators is 2.
? 4 : 5 :
9
2
= 8 : 10 : 9
The HCF of these 3 numbers is 1.
? 8 : 10 : 9 is the simplest form.
vi
2.5 : 6.5 : 8 = 25 : 65 : 80
The HCF of 25, 65 and 80 is 5.
? 25 : 65 : 80 =
25
65
80
=
25 ÷ 5
65 ÷ 5
80 ÷ 5
=
5
13
16
= 5 : 13 : 16
Q u e s t i o n : 2
Express each of the following ratios in simplest form:
i
75 paise : 3 rupees
ii
Page 2
Q u e s t i o n : 1
Express each of the following ratios in simplest form:
i
24 : 40
ii
13.5 : 15
iii
6
2
3
: 7
1
2
iv
1
6
:
1
9
v
4: 5:
9
2
vi
2.5 : 6.5 : 8
S o l u t i o n :
i
HCF of 24 and 40 is 8.
? 24 : 40 =
24
40
=
24 ÷ 8
40 ÷ 8
=
3
5
= 3 : 5
Hence, 24 : 40 in its simplest form is 3 : 5.
ii
HCF of 13.5 and 15 is 1.5.
13.5
15
=
135
150
The HCF of 135 and 150 is 15. =
135 ÷ 15
150 ÷ 15
=
9
10
Hence, 13.5 : 15 in its simplest form is 9 : 10.
iii
20
3
:
15
2
= 40 : 45
The HCF of 40 and 45 is 5.
? 40 : 45 =
40
45
=
40 ÷ 5
45 ÷ 5
=
8
9
= 8 : 9
Hence, 6
2
3
: 7
1
2
in its simplest form is 8 : 9
(iv) 9 : 6
The HCF of 9 and 6 is 3.
? 9 : 6 =
9
6
=
9 ÷ 3
6 ÷ 3
= 3 : 2
Hence,
1
6
:
1
9
in its simplest form is 3 : 2.
(v) LCM of the denominators is 2.
? 4 : 5 :
9
2
= 8 : 10 : 9
The HCF of these 3 numbers is 1.
? 8 : 10 : 9 is the simplest form.
vi
2.5 : 6.5 : 8 = 25 : 65 : 80
The HCF of 25, 65 and 80 is 5.
? 25 : 65 : 80 =
25
65
80
=
25 ÷ 5
65 ÷ 5
80 ÷ 5
=
5
13
16
= 5 : 13 : 16
Q u e s t i o n : 2
Express each of the following ratios in simplest form:
i
75 paise : 3 rupees
ii
1 m 5 cm : 63 cm
iii
1 hour 5 minutes : 45 minutes
iv
8 months : 1 year
v
2kg250g
: 3kg
vi
1 km : 750 m
S o l u t i o n :
i
Converting both the quantities into the same unit, we have:
75 paise : (3 ×
100) paise = 75 : 300
=
75
300
=
75 ÷ 75
300 ÷ 75
=
1
4
? HCFof75and300 = 75
= 1 paise : 4 paise
ii
Converting both the quantities into the same unit, we have:
105 cm : 63 cm =
105
63
=
105÷21
63÷21
=
5
3
? HCFof105and63 = 21
= 5 cm : 3 cm
iii
Converting both the quantities into the same unit
65 min : 45 min =
65
45
=
65÷5
45÷5
=
13
9
? HCFof65and45 = 5
= 13 min : 9 min
iv
Converting both the quantities into the same unit, we get:
8 months : 12 months =
8
12
=
8÷4
12÷4
=
2
3
? HCFof8and12 = 4
= 2 months : 3 months
v
Converting both the quantities into the same unit, we get:
2250g : 3000 g =
2250
3000
=
2250÷750
3000÷750
=
3
4
? HCFof2250and3000 = 750
= 3 g : 4 g
vi
Converting both the quantities into the same unit, we get:
1000 m : 750 m =
1000
750
=
1000÷250
750÷250
=
4
3
? HCFof1000and750 = 250
= 4 m : 3 m
Q u e s t i o n : 3
If A : B = 7 : 5 and B : C = 9 : 14, find A : C.
S o l u t i o n :
A
B
=
7
5
and
B
C
=
9
14
Therefore, we have:
A
B
×
B
C
=
7
5
×
9
14
A
C
=
9
10
? A : C = 9 : 10
Q u e s t i o n : 4
If A : B = 5 : 8 and B : C = 16 : 25, find A : C.
S o l u t i o n :
A
B
=
5
8
and
B
C
=
16
25
Now, we have:
A
B
×
B
C
=
5
8
×
16
25
?
A
C
=
2
5
? A : C = 2 : 5
Q u e s t i o n : 5
If A : B = 3 : 5 and B : C = 10 : 13, find A : B : C.
S o l u t i o n :
A : B = 3 : 5
Page 3
Q u e s t i o n : 1
Express each of the following ratios in simplest form:
i
24 : 40
ii
13.5 : 15
iii
6
2
3
: 7
1
2
iv
1
6
:
1
9
v
4: 5:
9
2
vi
2.5 : 6.5 : 8
S o l u t i o n :
i
HCF of 24 and 40 is 8.
? 24 : 40 =
24
40
=
24 ÷ 8
40 ÷ 8
=
3
5
= 3 : 5
Hence, 24 : 40 in its simplest form is 3 : 5.
ii
HCF of 13.5 and 15 is 1.5.
13.5
15
=
135
150
The HCF of 135 and 150 is 15. =
135 ÷ 15
150 ÷ 15
=
9
10
Hence, 13.5 : 15 in its simplest form is 9 : 10.
iii
20
3
:
15
2
= 40 : 45
The HCF of 40 and 45 is 5.
? 40 : 45 =
40
45
=
40 ÷ 5
45 ÷ 5
=
8
9
= 8 : 9
Hence, 6
2
3
: 7
1
2
in its simplest form is 8 : 9
(iv) 9 : 6
The HCF of 9 and 6 is 3.
? 9 : 6 =
9
6
=
9 ÷ 3
6 ÷ 3
= 3 : 2
Hence,
1
6
:
1
9
in its simplest form is 3 : 2.
(v) LCM of the denominators is 2.
? 4 : 5 :
9
2
= 8 : 10 : 9
The HCF of these 3 numbers is 1.
? 8 : 10 : 9 is the simplest form.
vi
2.5 : 6.5 : 8 = 25 : 65 : 80
The HCF of 25, 65 and 80 is 5.
? 25 : 65 : 80 =
25
65
80
=
25 ÷ 5
65 ÷ 5
80 ÷ 5
=
5
13
16
= 5 : 13 : 16
Q u e s t i o n : 2
Express each of the following ratios in simplest form:
i
75 paise : 3 rupees
ii
1 m 5 cm : 63 cm
iii
1 hour 5 minutes : 45 minutes
iv
8 months : 1 year
v
2kg250g
: 3kg
vi
1 km : 750 m
S o l u t i o n :
i
Converting both the quantities into the same unit, we have:
75 paise : (3 ×
100) paise = 75 : 300
=
75
300
=
75 ÷ 75
300 ÷ 75
=
1
4
? HCFof75and300 = 75
= 1 paise : 4 paise
ii
Converting both the quantities into the same unit, we have:
105 cm : 63 cm =
105
63
=
105÷21
63÷21
=
5
3
? HCFof105and63 = 21
= 5 cm : 3 cm
iii
Converting both the quantities into the same unit
65 min : 45 min =
65
45
=
65÷5
45÷5
=
13
9
? HCFof65and45 = 5
= 13 min : 9 min
iv
Converting both the quantities into the same unit, we get:
8 months : 12 months =
8
12
=
8÷4
12÷4
=
2
3
? HCFof8and12 = 4
= 2 months : 3 months
v
Converting both the quantities into the same unit, we get:
2250g : 3000 g =
2250
3000
=
2250÷750
3000÷750
=
3
4
? HCFof2250and3000 = 750
= 3 g : 4 g
vi
Converting both the quantities into the same unit, we get:
1000 m : 750 m =
1000
750
=
1000÷250
750÷250
=
4
3
? HCFof1000and750 = 250
= 4 m : 3 m
Q u e s t i o n : 3
If A : B = 7 : 5 and B : C = 9 : 14, find A : C.
S o l u t i o n :
A
B
=
7
5
and
B
C
=
9
14
Therefore, we have:
A
B
×
B
C
=
7
5
×
9
14
A
C
=
9
10
? A : C = 9 : 10
Q u e s t i o n : 4
If A : B = 5 : 8 and B : C = 16 : 25, find A : C.
S o l u t i o n :
A
B
=
5
8
and
B
C
=
16
25
Now, we have:
A
B
×
B
C
=
5
8
×
16
25
?
A
C
=
2
5
? A : C = 2 : 5
Q u e s t i o n : 5
If A : B = 3 : 5 and B : C = 10 : 13, find A : B : C.
S o l u t i o n :
A : B = 3 : 5
B : C = 10 : 13 =
10÷2
13÷2
= 5 :
13
2
Now, A : B : C = 3 : 5 :
13
2
? A : B : C = 6 : 10 : 13
Q u e s t i o n : 6
If A : B = 5 : 6 and B : C = 4 : 7, find A : B : C.
S o l u t i o n :
We have the following:
A : B = 5 : 6
B : C = 4 : 7 =
4
7
=
4×
6
4
7×
6
4
= 6 :
21
2
? A : B : C = 5 : 6 :
21
2
= 10 : 12 : 21
Q u e s t i o n : 7
Divide Rs 360 between Kunal and Mohit in the ratio 7 : 8.
S o l u t i o n :
Sum of the ratio terms = 7 + 8 = 15
Now, we have the following:
Kunal's share = Rs 360 ×
7
15
= 24 ×7
= Rs 168
Mohit's share = Rs 360 ×
8
15
= 24 ×8
= Rs 192
Q u e s t i o n : 8
Divide Rs 880 between Rajan and Kamal in the ratio
1
5
:
1
6
.
S o l u t i o n :
Sum of the ratio terms =
1
5
+
1
6
=
11
30
Now, we have the following:
Rajan's share = Rs 880 ×
1
5
11
30
= Rs 880 ×
6
11
= Rs 80 ×6
= Rs 480
Kamal's share = Rs 880 ×
1
6
11
30
= Rs 880 ×
5
11
= Rs 80 ×5
= Rs 400
Q u e s t i o n : 9
Divide Rs 5600 between A, B and C in the ratio 1 : 3 : 4.
S o l u t i o n :
Sum of the ratio terms is 1 +3 +4
= 8
We have the following:
A's share = Rs 5600 ×
1
8
= Rs
5600
8
= Rs 700
B's share = Rs 5600 ×
3
8
= Rs 700 × 3
= Rs 2100
C's share = Rs 5600 ×
4
8
= Rs 700 ×4
= Rs 2800
Q u e s t i o n : 1 0
What number must be added to each term to the ratio 9 : 16 to make the ratio 2 : 3?
S o l u t i o n :
Let x be the required number.
Then, (9 + x) : (16 + x) = 2 : 3
?
9+x
16+x
=
2
3
? 27 + 3x = 32 + 2x ? x = 5
Hence, 5 must be added to each term of the ratio 9 : 16 to make it 2 : 3.
Q u e s t i o n : 1 1
Page 4
Q u e s t i o n : 1
Express each of the following ratios in simplest form:
i
24 : 40
ii
13.5 : 15
iii
6
2
3
: 7
1
2
iv
1
6
:
1
9
v
4: 5:
9
2
vi
2.5 : 6.5 : 8
S o l u t i o n :
i
HCF of 24 and 40 is 8.
? 24 : 40 =
24
40
=
24 ÷ 8
40 ÷ 8
=
3
5
= 3 : 5
Hence, 24 : 40 in its simplest form is 3 : 5.
ii
HCF of 13.5 and 15 is 1.5.
13.5
15
=
135
150
The HCF of 135 and 150 is 15. =
135 ÷ 15
150 ÷ 15
=
9
10
Hence, 13.5 : 15 in its simplest form is 9 : 10.
iii
20
3
:
15
2
= 40 : 45
The HCF of 40 and 45 is 5.
? 40 : 45 =
40
45
=
40 ÷ 5
45 ÷ 5
=
8
9
= 8 : 9
Hence, 6
2
3
: 7
1
2
in its simplest form is 8 : 9
(iv) 9 : 6
The HCF of 9 and 6 is 3.
? 9 : 6 =
9
6
=
9 ÷ 3
6 ÷ 3
= 3 : 2
Hence,
1
6
:
1
9
in its simplest form is 3 : 2.
(v) LCM of the denominators is 2.
? 4 : 5 :
9
2
= 8 : 10 : 9
The HCF of these 3 numbers is 1.
? 8 : 10 : 9 is the simplest form.
vi
2.5 : 6.5 : 8 = 25 : 65 : 80
The HCF of 25, 65 and 80 is 5.
? 25 : 65 : 80 =
25
65
80
=
25 ÷ 5
65 ÷ 5
80 ÷ 5
=
5
13
16
= 5 : 13 : 16
Q u e s t i o n : 2
Express each of the following ratios in simplest form:
i
75 paise : 3 rupees
ii
1 m 5 cm : 63 cm
iii
1 hour 5 minutes : 45 minutes
iv
8 months : 1 year
v
2kg250g
: 3kg
vi
1 km : 750 m
S o l u t i o n :
i
Converting both the quantities into the same unit, we have:
75 paise : (3 ×
100) paise = 75 : 300
=
75
300
=
75 ÷ 75
300 ÷ 75
=
1
4
? HCFof75and300 = 75
= 1 paise : 4 paise
ii
Converting both the quantities into the same unit, we have:
105 cm : 63 cm =
105
63
=
105÷21
63÷21
=
5
3
? HCFof105and63 = 21
= 5 cm : 3 cm
iii
Converting both the quantities into the same unit
65 min : 45 min =
65
45
=
65÷5
45÷5
=
13
9
? HCFof65and45 = 5
= 13 min : 9 min
iv
Converting both the quantities into the same unit, we get:
8 months : 12 months =
8
12
=
8÷4
12÷4
=
2
3
? HCFof8and12 = 4
= 2 months : 3 months
v
Converting both the quantities into the same unit, we get:
2250g : 3000 g =
2250
3000
=
2250÷750
3000÷750
=
3
4
? HCFof2250and3000 = 750
= 3 g : 4 g
vi
Converting both the quantities into the same unit, we get:
1000 m : 750 m =
1000
750
=
1000÷250
750÷250
=
4
3
? HCFof1000and750 = 250
= 4 m : 3 m
Q u e s t i o n : 3
If A : B = 7 : 5 and B : C = 9 : 14, find A : C.
S o l u t i o n :
A
B
=
7
5
and
B
C
=
9
14
Therefore, we have:
A
B
×
B
C
=
7
5
×
9
14
A
C
=
9
10
? A : C = 9 : 10
Q u e s t i o n : 4
If A : B = 5 : 8 and B : C = 16 : 25, find A : C.
S o l u t i o n :
A
B
=
5
8
and
B
C
=
16
25
Now, we have:
A
B
×
B
C
=
5
8
×
16
25
?
A
C
=
2
5
? A : C = 2 : 5
Q u e s t i o n : 5
If A : B = 3 : 5 and B : C = 10 : 13, find A : B : C.
S o l u t i o n :
A : B = 3 : 5
B : C = 10 : 13 =
10÷2
13÷2
= 5 :
13
2
Now, A : B : C = 3 : 5 :
13
2
? A : B : C = 6 : 10 : 13
Q u e s t i o n : 6
If A : B = 5 : 6 and B : C = 4 : 7, find A : B : C.
S o l u t i o n :
We have the following:
A : B = 5 : 6
B : C = 4 : 7 =
4
7
=
4×
6
4
7×
6
4
= 6 :
21
2
? A : B : C = 5 : 6 :
21
2
= 10 : 12 : 21
Q u e s t i o n : 7
Divide Rs 360 between Kunal and Mohit in the ratio 7 : 8.
S o l u t i o n :
Sum of the ratio terms = 7 + 8 = 15
Now, we have the following:
Kunal's share = Rs 360 ×
7
15
= 24 ×7
= Rs 168
Mohit's share = Rs 360 ×
8
15
= 24 ×8
= Rs 192
Q u e s t i o n : 8
Divide Rs 880 between Rajan and Kamal in the ratio
1
5
:
1
6
.
S o l u t i o n :
Sum of the ratio terms =
1
5
+
1
6
=
11
30
Now, we have the following:
Rajan's share = Rs 880 ×
1
5
11
30
= Rs 880 ×
6
11
= Rs 80 ×6
= Rs 480
Kamal's share = Rs 880 ×
1
6
11
30
= Rs 880 ×
5
11
= Rs 80 ×5
= Rs 400
Q u e s t i o n : 9
Divide Rs 5600 between A, B and C in the ratio 1 : 3 : 4.
S o l u t i o n :
Sum of the ratio terms is 1 +3 +4
= 8
We have the following:
A's share = Rs 5600 ×
1
8
= Rs
5600
8
= Rs 700
B's share = Rs 5600 ×
3
8
= Rs 700 × 3
= Rs 2100
C's share = Rs 5600 ×
4
8
= Rs 700 ×4
= Rs 2800
Q u e s t i o n : 1 0
What number must be added to each term to the ratio 9 : 16 to make the ratio 2 : 3?
S o l u t i o n :
Let x be the required number.
Then, (9 + x) : (16 + x) = 2 : 3
?
9+x
16+x
=
2
3
? 27 + 3x = 32 + 2x ? x = 5
Hence, 5 must be added to each term of the ratio 9 : 16 to make it 2 : 3.
Q u e s t i o n : 1 1
What number must be subtracted from each term of ratio 17 : 33 so that the ratio becomes 7 : 15?
S o l u t i o n :
Suppose that x is the number that must be subtracted.
Then, (17 - x) : (33 - x) = 7 : 15
?
17 - x
33 - x
=
7
15
? 255 - 15x = 231 - 7x ? 8x = 255 - 231 = 24 ? x = 3
Hence, 3 must be subtracted from each term of ratio 17 : 33 so that it becomes 7 : 15.
Q u e s t i o n : 1 2
Two numbers are in the ratio 7 : 11. If added to each of the numbers, the ratio becomes 2 : 3. Find the numbers.
S o l u t i o n :
Suppose that the numbers are 7x and 11x.
Then, (7x + 7) : (11x + 7) = 2 : 3
?
7x + 7
11x + 7
=
2
3
? 21x + 21 = 22x + 14
? x = 7
Hence, the numbers are (7 ×
7 =) 49 and (11 ×
7 =) 77.
Q u e s t i o n : 1 3
Two numbers are in the ratio 5 : 9. On subtracting 3 from each, the ratio becomes 1 : 2. Find the numbers.
S o l u t i o n :
Suppose that the numbers are 5x and 9x.
Then, (5x - 3) : (9x - 3) = 1 : 2
?
5x - 3
9x -3
=
1
2
? 10x - 6 = 9x - 3
? x = 3
Hence, the numbers are (5 ×
3 =) 15 and (9 ×
3 =) 27.
Q u e s t i o n : 1 4
Two numbers are in the ratio 3 : 4. If their LCM is 180, find the numbers.
S o l u t i o n :
Let the numbers be 3x and 4x.
Their LCM is 12x.
Then, 12x = 180
? x = 15
? The numbers are (3 ×
15 =) 45 and (4 ×
15 =) 60.
Q u e s t i o n : 1 5
The ages of A and B are in the ratio 8 : 3. Six years hence, their ages will be in the ratio 9 : 4. Find their present ages.
S o l u t i o n :
Suppose that the present ages of A and B are 8x yrs and 3x yrs.
Then, (8x + 6) : (3x + 6) = 9 : 4
?
8x+6
3x+6
=
9
4
? 32x + 24 = 27x + 54
? 5x = 30
? x = 6
Now, present age of A = 8 ×
6 yrs = 48 yrs
Present age of B = 3 ×
6 yrs = 18 yrs
Q u e s t i o n : 1 6
The ratio of copper and zinc in an alloy is 9 : 5. If the weight of copper in the alloy is 48.6 grams, find the weight of zinc in the alloy.
S o l u t i o n :
Suppose that the weight of zinc is x g.
Then, 48.6 : x = 9 : 5
? x =
48.6×5
9
=
243
9
= 27
Hence, the weight of zinc in the alloy is 27 g.
Q u e s t i o n : 1 7
The ratio of boys and girls in a school is 8 : 3. If the total number of girls be 375, find the number of boys in the school.
Page 5
Q u e s t i o n : 1
Express each of the following ratios in simplest form:
i
24 : 40
ii
13.5 : 15
iii
6
2
3
: 7
1
2
iv
1
6
:
1
9
v
4: 5:
9
2
vi
2.5 : 6.5 : 8
S o l u t i o n :
i
HCF of 24 and 40 is 8.
? 24 : 40 =
24
40
=
24 ÷ 8
40 ÷ 8
=
3
5
= 3 : 5
Hence, 24 : 40 in its simplest form is 3 : 5.
ii
HCF of 13.5 and 15 is 1.5.
13.5
15
=
135
150
The HCF of 135 and 150 is 15. =
135 ÷ 15
150 ÷ 15
=
9
10
Hence, 13.5 : 15 in its simplest form is 9 : 10.
iii
20
3
:
15
2
= 40 : 45
The HCF of 40 and 45 is 5.
? 40 : 45 =
40
45
=
40 ÷ 5
45 ÷ 5
=
8
9
= 8 : 9
Hence, 6
2
3
: 7
1
2
in its simplest form is 8 : 9
(iv) 9 : 6
The HCF of 9 and 6 is 3.
? 9 : 6 =
9
6
=
9 ÷ 3
6 ÷ 3
= 3 : 2
Hence,
1
6
:
1
9
in its simplest form is 3 : 2.
(v) LCM of the denominators is 2.
? 4 : 5 :
9
2
= 8 : 10 : 9
The HCF of these 3 numbers is 1.
? 8 : 10 : 9 is the simplest form.
vi
2.5 : 6.5 : 8 = 25 : 65 : 80
The HCF of 25, 65 and 80 is 5.
? 25 : 65 : 80 =
25
65
80
=
25 ÷ 5
65 ÷ 5
80 ÷ 5
=
5
13
16
= 5 : 13 : 16
Q u e s t i o n : 2
Express each of the following ratios in simplest form:
i
75 paise : 3 rupees
ii
1 m 5 cm : 63 cm
iii
1 hour 5 minutes : 45 minutes
iv
8 months : 1 year
v
2kg250g
: 3kg
vi
1 km : 750 m
S o l u t i o n :
i
Converting both the quantities into the same unit, we have:
75 paise : (3 ×
100) paise = 75 : 300
=
75
300
=
75 ÷ 75
300 ÷ 75
=
1
4
? HCFof75and300 = 75
= 1 paise : 4 paise
ii
Converting both the quantities into the same unit, we have:
105 cm : 63 cm =
105
63
=
105÷21
63÷21
=
5
3
? HCFof105and63 = 21
= 5 cm : 3 cm
iii
Converting both the quantities into the same unit
65 min : 45 min =
65
45
=
65÷5
45÷5
=
13
9
? HCFof65and45 = 5
= 13 min : 9 min
iv
Converting both the quantities into the same unit, we get:
8 months : 12 months =
8
12
=
8÷4
12÷4
=
2
3
? HCFof8and12 = 4
= 2 months : 3 months
v
Converting both the quantities into the same unit, we get:
2250g : 3000 g =
2250
3000
=
2250÷750
3000÷750
=
3
4
? HCFof2250and3000 = 750
= 3 g : 4 g
vi
Converting both the quantities into the same unit, we get:
1000 m : 750 m =
1000
750
=
1000÷250
750÷250
=
4
3
? HCFof1000and750 = 250
= 4 m : 3 m
Q u e s t i o n : 3
If A : B = 7 : 5 and B : C = 9 : 14, find A : C.
S o l u t i o n :
A
B
=
7
5
and
B
C
=
9
14
Therefore, we have:
A
B
×
B
C
=
7
5
×
9
14
A
C
=
9
10
? A : C = 9 : 10
Q u e s t i o n : 4
If A : B = 5 : 8 and B : C = 16 : 25, find A : C.
S o l u t i o n :
A
B
=
5
8
and
B
C
=
16
25
Now, we have:
A
B
×
B
C
=
5
8
×
16
25
?
A
C
=
2
5
? A : C = 2 : 5
Q u e s t i o n : 5
If A : B = 3 : 5 and B : C = 10 : 13, find A : B : C.
S o l u t i o n :
A : B = 3 : 5
B : C = 10 : 13 =
10÷2
13÷2
= 5 :
13
2
Now, A : B : C = 3 : 5 :
13
2
? A : B : C = 6 : 10 : 13
Q u e s t i o n : 6
If A : B = 5 : 6 and B : C = 4 : 7, find A : B : C.
S o l u t i o n :
We have the following:
A : B = 5 : 6
B : C = 4 : 7 =
4
7
=
4×
6
4
7×
6
4
= 6 :
21
2
? A : B : C = 5 : 6 :
21
2
= 10 : 12 : 21
Q u e s t i o n : 7
Divide Rs 360 between Kunal and Mohit in the ratio 7 : 8.
S o l u t i o n :
Sum of the ratio terms = 7 + 8 = 15
Now, we have the following:
Kunal's share = Rs 360 ×
7
15
= 24 ×7
= Rs 168
Mohit's share = Rs 360 ×
8
15
= 24 ×8
= Rs 192
Q u e s t i o n : 8
Divide Rs 880 between Rajan and Kamal in the ratio
1
5
:
1
6
.
S o l u t i o n :
Sum of the ratio terms =
1
5
+
1
6
=
11
30
Now, we have the following:
Rajan's share = Rs 880 ×
1
5
11
30
= Rs 880 ×
6
11
= Rs 80 ×6
= Rs 480
Kamal's share = Rs 880 ×
1
6
11
30
= Rs 880 ×
5
11
= Rs 80 ×5
= Rs 400
Q u e s t i o n : 9
Divide Rs 5600 between A, B and C in the ratio 1 : 3 : 4.
S o l u t i o n :
Sum of the ratio terms is 1 +3 +4
= 8
We have the following:
A's share = Rs 5600 ×
1
8
= Rs
5600
8
= Rs 700
B's share = Rs 5600 ×
3
8
= Rs 700 × 3
= Rs 2100
C's share = Rs 5600 ×
4
8
= Rs 700 ×4
= Rs 2800
Q u e s t i o n : 1 0
What number must be added to each term to the ratio 9 : 16 to make the ratio 2 : 3?
S o l u t i o n :
Let x be the required number.
Then, (9 + x) : (16 + x) = 2 : 3
?
9+x
16+x
=
2
3
? 27 + 3x = 32 + 2x ? x = 5
Hence, 5 must be added to each term of the ratio 9 : 16 to make it 2 : 3.
Q u e s t i o n : 1 1
What number must be subtracted from each term of ratio 17 : 33 so that the ratio becomes 7 : 15?
S o l u t i o n :
Suppose that x is the number that must be subtracted.
Then, (17 - x) : (33 - x) = 7 : 15
?
17 - x
33 - x
=
7
15
? 255 - 15x = 231 - 7x ? 8x = 255 - 231 = 24 ? x = 3
Hence, 3 must be subtracted from each term of ratio 17 : 33 so that it becomes 7 : 15.
Q u e s t i o n : 1 2
Two numbers are in the ratio 7 : 11. If added to each of the numbers, the ratio becomes 2 : 3. Find the numbers.
S o l u t i o n :
Suppose that the numbers are 7x and 11x.
Then, (7x + 7) : (11x + 7) = 2 : 3
?
7x + 7
11x + 7
=
2
3
? 21x + 21 = 22x + 14
? x = 7
Hence, the numbers are (7 ×
7 =) 49 and (11 ×
7 =) 77.
Q u e s t i o n : 1 3
Two numbers are in the ratio 5 : 9. On subtracting 3 from each, the ratio becomes 1 : 2. Find the numbers.
S o l u t i o n :
Suppose that the numbers are 5x and 9x.
Then, (5x - 3) : (9x - 3) = 1 : 2
?
5x - 3
9x -3
=
1
2
? 10x - 6 = 9x - 3
? x = 3
Hence, the numbers are (5 ×
3 =) 15 and (9 ×
3 =) 27.
Q u e s t i o n : 1 4
Two numbers are in the ratio 3 : 4. If their LCM is 180, find the numbers.
S o l u t i o n :
Let the numbers be 3x and 4x.
Their LCM is 12x.
Then, 12x = 180
? x = 15
? The numbers are (3 ×
15 =) 45 and (4 ×
15 =) 60.
Q u e s t i o n : 1 5
The ages of A and B are in the ratio 8 : 3. Six years hence, their ages will be in the ratio 9 : 4. Find their present ages.
S o l u t i o n :
Suppose that the present ages of A and B are 8x yrs and 3x yrs.
Then, (8x + 6) : (3x + 6) = 9 : 4
?
8x+6
3x+6
=
9
4
? 32x + 24 = 27x + 54
? 5x = 30
? x = 6
Now, present age of A = 8 ×
6 yrs = 48 yrs
Present age of B = 3 ×
6 yrs = 18 yrs
Q u e s t i o n : 1 6
The ratio of copper and zinc in an alloy is 9 : 5. If the weight of copper in the alloy is 48.6 grams, find the weight of zinc in the alloy.
S o l u t i o n :
Suppose that the weight of zinc is x g.
Then, 48.6 : x = 9 : 5
? x =
48.6×5
9
=
243
9
= 27
Hence, the weight of zinc in the alloy is 27 g.
Q u e s t i o n : 1 7
The ratio of boys and girls in a school is 8 : 3. If the total number of girls be 375, find the number of boys in the school.
S o l u t i o n :
Suppose that the number of boys is x.
Then, x : 375 = 8 : 3
? x =
8×375
3
= 8 ×125
= 1000
Hence, the number of girls in the school is 1000.
Q u e s t i o n : 1 8
The ratio of monthly income to the savings of a family is 11 : 2. If the savings be Rs 2500, find the income and expenditure.
S o l u t i o n :
Suppose that the monthly income of the family is Rs x.
Then, x : 2500 = 11 : 2
? x =
11×2500
2
= 11 ×1250
? x = Rs 13750
Hence, the income is Rs 13,750.
? Expenditure = monthlyincome -savings
=Rs 13750 -2500
= Rs 11250
Q u e s t i o n : 1 9
A bag contains Rs 750 in the form of rupee, 50 P and 25 P coins in the ratio 5 : 8 : 4. Find the number of coins of each type.
S o l u t i o n :
Let the numbers one rupee, fifty paise and twenty-five paise coins be 5x, 8x and 4x, respectively.
Total value of these coins = (5x ×
100
100
+ 8x ×
50
100
+ 4x ×
25
100
)
? 5x +
8x
2
+
4x
4
=
20x + 16x + 4x
4
=
40x
4
= 10x
However, the total value is Rs 750.
? 750 = 10x
? x = 75
Hence, number of one rupee coins = 5 ×
75 = 375
Number of fifty paise coins = 8 ×
75 = 600
Number of twenty-five paise coins = 4 ×
75 = 300
Q u e s t i o n : 2 0
If (4x + 5) : (3x + 11) = 13 : 17, find the value of x.
S o l u t i o n :
(4x + 5) : (3x + 11) = 13 : 17
?
4x+ 5
3x + 11
=
13
17
? 68x + 85 = 39x + 143 ? 29x = 58 ? x = 2
Q u e s t i o n : 2 1
If x : y = 3 : 4, find (3x + 4y) : (5x + 6y).
S o l u t i o n :
x
y
=
3
4
? x =
3y
4
Now, we have (3x + 4y) : (5x + 6y)
=
3x +4y
5x + 6y
=
3×
3y
4
+4y
5×
3y
4
+6y
=
9y+16y
15y +24y
=
25y
39y
=
25
39
= 25 : 39
Q u e s t i o n : 2 2
If x : y = 6 : 11, find (8x - 3y) : (3x + 2y).
S o l u t i o n :
x
y
=
6
11
? x =
6y
11
Now, we have:
8x -3y
3x + 2y
=
8×
6y
11
-3y
3×
6y
11
+2y
=
48y-33y
18y + 22y
=
15y
40y
=
3
8
? (8x - 3y) : (3x + 2y) = 3 : 8
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