Step-by-step explanation:

ratio of number = 4:5:6

then the no. be 4x, 5x and 6x

a/q

6x+4x= 5x+55

10x-5x= 55

5x= 55

x= 11

hence the no. be 4x= 44

5x= 55

6x= 66

2)

let the no. be x

4/5 of x = 4x/5

2/3 of no. = 2x/3

their differences= 10

a/q

4x/5 - 2x/3 = 10

(12x-10x)/15= 10

2x/15= 10

2x= 150

x= 75

#666

Verified answer

Answer:

[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \:(1) \: \: Numbers\:are\:44, 55 \: and \: 66\qquad \: \\ \\& \qquad \:\sf \: (2) \: \: Number\:is \: 75 \end{aligned}} \qquad \: \\ \\ [/tex]

Step-by-step explanation:

[tex]\large\underline{\sf{Solution-1}}[/tex]

Given that three numbers are in the ratio 4 : 5 : 6.

Let assume that numbers be 4x, 5x, 6x.

According to statement, the sum of the largest and the smallest equals the sum of the third and 55.

So, it means

[tex]\sf \: 6x + 4x = 5x + 55 \\ \\ [/tex]

[tex]\sf \: 10x = 5x + 55 \\ \\ [/tex]

[tex]\sf \: 10x - 5x = 55 \\ \\ [/tex]

[tex]\sf \: 5x = 55 \\ \\ [/tex]

[tex]\sf\implies \sf \: x = 11 \\ \\ [/tex]

Thus,

First number = 4x = 4 × 11 = 44

Second number = 5x = 5 × 11 = 55

Third numbers = 6x = 6 × 11 = 66

Hence,

[tex]\sf\implies \sf \: Numbers\:are\:44, 55 \: and \: 66 \\ \\ [/tex]

[tex]\rule{190pt}{1pt}[/tex]

[tex]\large\underline{\sf{Solution-2}}[/tex]

Let number be x.

According to statement, four- fifths of a number is 10 more than two-thirds of the number.

[tex]\sf \: \dfrac{4x}{5} = 10 + \dfrac{2x}{3} \\ \\ [/tex]

[tex]\sf \: \dfrac{4x}{5} - \dfrac{2x}{3} = 10 \\ \\ [/tex]

[tex]\sf \: \dfrac{12x - 10x}{15}= 10 \\ \\ [/tex]

[tex]\sf \: \dfrac{2x}{15}= 10 \\ \\ [/tex]

[tex]\sf \:x = \dfrac{15}{2} \times 10 \\ \\ [/tex]

[tex]\sf \:x = 15 \times 5 \\ \\ [/tex]

[tex]\sf\implies \sf \:x = 75 \\ \\ [/tex]

Hence,

[tex]\sf\implies \sf \:Number\:is \: 75 \: \\ \\ [/tex]

Thus,

[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \:(1) \: \: Numbers\:are\:44, 55 \: and \: 66\qquad \: \\ \\& \qquad \:\sf \: (2) \: \: Number\:is \: 75 \end{aligned}} \qquad \: \\ \\ [/tex]