What number must be subtracted from each term of the ratio 19 21 to make it 7 8

What number must be subtracted from each of the numbers $10,12,19,24$ to get the numbers which are in proportion?

Answer

Hint: The number subtracting from the given numbers is the same. So we can form an equation using the definition of proportion. Proportionality can be expressed using fractions. Solving this we get the answer.

Formula used: If we say four numbers $a,b,c,d$ are in proportion we mean $a:b = c:d$ or $\dfrac{a}{b} = \dfrac{c}{d}$.

Complete step-by-step solution:
We are given four numbers $10,12,19,24$.
It is said that the same number when subtracted from these numbers make a proportion.
If we say four numbers $a,b,c,d$ are in proportion we mean $a:b = c:d$ or $\dfrac{a}{b} = \dfrac{c}{d}$.
Let the number subtracting be $x$.
So we have,
$\Rightarrow$$\dfrac{{10 - x}}{{12 - x}} = \dfrac{{19 - x}}{{24 - x}}$
Cross multiplying we have,
$\Rightarrow$$[10 - x][24 - x] = [12 - x][19 - x]$
Opening the brackets we get,
$\Rightarrow$$240 - 10x - 24x + {x^2} = 228 - 12x - 19x + {x^2}$
Simplifying we get,
$\Rightarrow$$240 - 34x + {x^2} = 228 - 31x + {x^2}$
Cancelling ${x^2}$ from both sides we get,
$\Rightarrow$$240 - 34x = 228 - 31x$
Rearranging the terms we have,
$\Rightarrow$$240 - 228 = - 31x + 34x$
$ \Rightarrow 3x = 12$
Dividing both sides by $3$ we get,
$\Rightarrow$$x = \dfrac{{12}}{3} = 4$
That is, $4$ has to be subtracted from the numbers to make them proportion.

$\therefore $ The answer is $4$.

Note: Proportion says that two ratios [or fractions] are equal.
If $a:b = c:d$, then the quantity $d$ is called the fourth proportional to $a,b$ and $c$.
Ratio and proportion is a widely used concept in Mathematics as well as in day to day life.
If the number subtracting from each number is different we cannot form an equation like this. So we need more information for those problems. But here this works, since the same number is subtracting from each. So we got an equation with a single variable which could be solved easily.

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Page No 124:

Question 1:

Express each of the following ratios in simplest form:

[i] 24 : 40
[ii] 13.5 : 15
[iii] 623:712
[iv] 16:19
[v] 4:5:92
[vi] 2.5 : 6.5 : 8

Answer:

[i] HCF of 24 and 40 is 8.
∴ 24 : 40 = 2440 = 24 ÷ 840 ÷ 8= 35 = 3 : 5

Hence, 24 : 40 in its simplest form is 3 : 5.

[ii] HCF of 13.5 and 15 is 1.5.

 13.515= 135150The HCF of 135 and 150 is 15.=135 ÷ 15150 ÷ 15 =910

Hence, 13.5 : 15 in its simplest form is 9 : 10.

[iii]  203 : 152=40 : 45
The HCF of 40 and 45 is 5.

∴ 40 : 45 = 4045 =40 ÷ 545 ÷ 5=89  = 8 : 9

Hence, 623 : 712 in its simplest form is 8 : 9

[iv] 9 : 6
The HCF of 9 and 6 is 3.

[vi] 2.5 : 6.5 : 8 = 25 : 65 : 80

The HCF of 25, 65 and 80 is 5.
∴ 25 : 65 : 80 = 256580= 25 ÷ 565  ÷ 580 ÷ 5=51316 = 5 : 13 : 16

Page No 124:

Question 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] [2 kg 250 g] : [3 kg]
[vi] 1 km : 750 m

Answer:

[i] Converting both the quantities into the same unit, we have:
   75 paise : [3 × 100] paise = 75 : 300

= 75300= 75 ÷  75300 ÷ 75=14    [∵ HCF of 75 and 300 = 75]
= 1 paise : 4 paise

[ii]  Converting both the quantities into the same unit, we have:
105 cm : 63 cm = 10563=105÷2163 ÷21= 53   [∵ HCF of 105 and 63 = 21]
= 5 cm : 3 cm

[iii] Converting both the quantities into the same unit
65 min : 45 min = 6545= 65÷545÷5=139   [∵ HCF of 65 and 45 = 5]
= 13 min : 9 min

[iv] Converting both the quantities into the same unit, we get:
8 months : 12 months = 812=8÷412÷4=23  [∵ HCF of 8 and 12 = 4]
= 2 months : 3 months

[v] Converting both the quantities into the same unit, we get:

2250g : 3000 g = 22503000=2250÷7503000÷750= 34    [∵ HCF of 2250 and 3000 = 750]

= 3 g : 4 g

[vi]  Converting both the quantities into the same unit, we get:
1000 m : 750 m =  1000750=1000÷250750÷250 = 43    [∵ HCF of 1000 and 750 = 250]
= 4 m : 3 m

Page No 124:

Question 3:

If A : B = 7 : 5 and B : C = 9 : 14, find A : C.

Answer:

AB  = 75 and  BC = 914

Therefore, we have:

AB×BC = 75×914AC = 910

∴ A : C = 9 : 10

Page No 124:

Question 4:

If A : B = 5 : 8 and B : C = 16 : 25, find A : C.

Answer:

AB=58 and BC = 1625Now,  we have:AB×BC =  58×1625⇒AC = 25

∴ A : C = 2 : 5

Page No 124:

Question 5:

If A : B = 3 : 5 and B : C = 10 : 13, find A : B : C.

Answer:

A : B = 3 : 5

B : C = 10 : 13 =  10÷213÷2 =5 :132

Now, A : B : C = 3 : 5 : 132

∴ A : B : C = 6 : 10 : 13

Page No 124:

Question 6:

If A : B = 5 : 6 and B : C = 4 : 7, find A : B : C.

Answer:

We have the following:

A : B = 5 : 6
B : C = 4 : 7  = 47 = 4×647×64=  6 : 212

∴ A : B : C =  5 : 6 : 212 =  10 : 12 : 21

Page No 124:

Question 7:

Divide Rs 360 between Kunal and Mohit in the ratio 7 : 8.

Answer:

Sum of the ratio terms  = 7 + 8 = 15

Now, we have the following:

Kunal's share = Rs 360 ×715= 24×7 = Rs 168

Mohit's share = Rs 360 ×815 = 24×8 = Rs 192

Page No 125:

Question 8:

Divide Rs 880 between Rajan and Kamal in the ratio 15:16.

Answer:

Sum of the ratio terms = 15+16=1130

Now, we have the following:
Rajan's share = Rs 880 ×151130 = Rs 880 ×611 = Rs 80×6   =  Rs 480
Kamal's share = Rs 880 ×161130= Rs 880 ×511= Rs 80 ×5 = Rs 400

Page No 125:

Question 9:

Divide Rs 5600 between A, B and C in the ratio 1 : 3 : 4.

Answer:

Sum of the ratio terms is [1 + 3 + 4] = 8

We have the following:

A's share =  Rs 5600 ×18 =Rs 56008  = Rs 700

B's share =  Rs 5600 ×38= Rs 700 × 3 = Rs 2100

C's share = Rs 5600 ×48 =Rs  700 ×4 = Rs 2800

Page No 125:

Question 10:

What number must be added to each term to the ratio 9 : 16 to make the ratio 2 : 3?

Answer:

Let x be the required number.
Then, [9 + x] : [16 + x] = 2 : 3

⇒9+x16+ x = 23⇒27 + 3x = 32 + 2x⇒x =5

Hence, 5 must be added to each term of the ratio 9 : 16 to make it 2 : 3.

Page No 125:

Question 11:

What number must be subtracted from each term of ratio 17 : 33 so that the ratio becomes 7 : 15?

Answer:

Suppose that x is the number that must be subtracted.
Then, [17 − x] : [33 − x] = 7 : 15

⇒17 - x33 - x=715⇒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.

Page No 125:

Question 12:

Two numbers are in the ratio 7 : 11. If added to each of the numbers, the ratio becomes 2 : 3. Find the numbers.

Answer:

Suppose that the numbers are 7x and 11x.

Then, [7x + 7] : [11x + 7] = 2 : 3
⇒ 7x  + 711x + 7=23

⇒ 21x + 21 = 22x + 14

⇒ x = 7

Hence, the numbers are [7 × 7 =] 49 and [11 × 7 =] 77.

Page No 125:

Question 13:

Two numbers are in the ratio 5 : 9. On subtracting 3 from each, the ratio becomes 1 : 2. Find the numbers.

Answer:

Suppose that the numbers are 5x and 9x.
Then, [5x − 3] : [9x − 3] = 1 : 2

⇒ 5x - 3 9x -3=12

⇒ 10x − 6 = 9x− 3
⇒ x = 3

Hence, the numbers are [5 × 3 =] 15 and [9 × 3 =] 27.

Page No 125:

Question 14:

Two numbers are in the ratio 3 : 4. If their LCM is 180, find the numbers.

Answer:

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.

Page No 125:

Question 15:

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.

Answer:

Suppose that the present ages of A and B are 8x yrs and 3x yrs.
Then, [8x  + 6] : [3x + 6] = 9 : 4
⇒ 8x+63x+6=  94
⇒ 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

Page No 125:

Question 16:

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.

Answer:

Suppose that the weight of zinc is x g.

Then, 48.6 : x = 9 : 5

⇒ x = 48.6×59=2439 = 27

Hence, the weight of zinc in the alloy is 27 g.

Page No 125:

Question 17:

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.

Answer:

Suppose that the number of boys is x.
Then, x : 375 = 8 : 3

⇒ x = 8×3753=8×125 = 1000

Hence, the number of girls in the school is 1000.

Page No 125:

Question 18:

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.

Answer:

Suppose that the monthly income of the family is Rs x.

Then, x : 2500 = 11 : 2

⇒ x = 11× 25002=11×1250
⇒ x = Rs 13750

Hence, the income is Rs 13,750.
∴ Expenditure = [monthly income − savings]
                         =Rs [13750 − 2500]
                          = Rs 11250

Page No 125:

Question 19:

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.

Answer:

Let the numbers one rupee, fifty paise and twenty-five paise coins be 5x, 8x and 4x, respectively.

Total value of these coins = [5x  ×100100+ 8x×50100 + 4x ×25100]

 ⇒5x + 8x2 +  4x4= 20x + 16x + 4x4=40x4=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

Page No 125:

Question 20:

If [4x + 5] : [3x + 11] = 13 : 17, find the value of x.

Answer:

[4x + 5] : [3x + 11] = 13 : 17

⇒4x+ 53x + 11=1317⇒68x + 85 = 39x  + 143⇒29x =  58⇒x = 2

Page No 125:

Question 21:

If x : y = 3 : 4, find [3x + 4y] : [5x + 6y].

Answer:

xy = 34⇒x=3y4

Now, we have [3x + 4y] : [5x + 6y]
=3x +4y5x + 6y=3×3y4+4y 5×3y4+6y= 9y+16y15y +24y  =  25y39y=2539

= 25 : 39

Page No 125:

Question 22:

If x : y = 6 : 11, find [8x − 3y] : [3x + 2y].

Answer:

xy =  611⇒x = 6y11

Now, we have:

8x -3y3x + 2y = 8×6y11 -3y3×6y11+2y=48y-33 y18y + 22y =15y40y=38

∴ [8x − 3y] : [3x + 2y] = 3 : 8

Page No 125:

Question 23:

Two numbers are in the ratio 5 : 7. If the sum of the numbers is 720, find the numbers.

Answer:

Suppose that the numbers are 5x and 7x.
The sum of the numbers is 720.
i.e., 5x + 7x = 720
⇒ 12x= 720
⇒ x = 60

Hence, the numbers are [5 × 60 =] 300 and [7 × 60 =] 420.

Page No 125:

Question 24:

Which ratio is greater?

[i] [5 : 6] or [7 : 9]
[ii] [2 : 3] or [4 : 7]
[iii] [1 : 2] or [4 : 7]
[iv] [3 : 5] or [8 : 13]

Answer:

[i] The LCM of 6 and 9 is 18.

56=5×36×3 =151879=7×29×2=1418Clearly , 1418