Question

Divide 784 into four parts such that 4 times the first part, 3 times the second part, twice the third part are each equal to 12 times the fourth part. find the first part

Answer 1

Given:

• The number $784$ is divided into $4$ parts.
• $4times$ the first part is equal to $12 times$ the fourth part.
• $3times$ the second part is equal to $12 times$ the fourth part.
• $2times$ the third part is equal to $12 times$ the fourth part.

To find:

The value of the first part.

Concept to be used:

Write the first three parts in terms of the fourth part and the equate the sum all the four parts, so obtained, to $784$. Solve for the fourth part and substitute that accordingly to find the first part.

Solution:

• Let the four parts be denoted as $x_1, x_2, x_3$ and $x_4$.
• As per the given data, $4times$ the first part is equal to $12 times$ the fourth part. This can be represented as follows:

⇒$4x_1=12x_4$

⇒$x_1=3x_4$.......(1)

• $3times$ the second part is equal to $12 times$ the fourth part. This is represented as follows:

$3x_2=12x_4\\x_2=4x_4$........(2)

• Also, given that, $2times$ the third part is equal to $12 times$ the fourth part.

⇒$2x_3=12x_4$

⇒$x_3=6x_4$........(3)

• The first three parts are written in terms of the fourth part which are represented in equations (1), (2) and (3).
• Now, from the given data, the sum of four parts is equal to $784$. This means:

$x_1+x_2+x_3+x_4=784$........(4)

• Substitute the equations (1), (2) and (3) in the equation (4). We get:

$3x_4+4x_4+6x_4+x_4=784\\14x_4=784\\x_4=56$

⇒The value of $x_4$ is $56$.

• Now, substitute the value of $x_4$ in the equation (1).

$x_1=3(x_4)\\x_1=3(56)\\x_1=168$

Final Answer:

The required value of the first part is $168$.

Answer 2

Answer:

Given:

The number \begin{gathered}784\\\end{gathered}

784

is divided into \begin{gathered}4\\\end{gathered}

4

parts.

\begin{gathered}4times\\\end{gathered}

4times

the first part is equal to \begin{gathered}12 times\\\end{gathered}

12times

the fourth part.

\begin{gathered}3times\\\end{gathered}

3times

the second part is equal to \begin{gathered}12 times\\\end{gathered}

12times

the fourth part.

\begin{gathered}2times\\\end{gathered}

2times

the third part is equal to \begin{gathered}12 times\\\end{gathered}

12times

the fourth part.

To find:

The value of the first part.

Concept to be used:

Write the first three parts in terms of the fourth part and the equate the sum all the four parts, so obtained, to \begin{gathered}784\\\end{gathered}

784

. Solve for the fourth part and substitute that accordingly to find the first part.

Solution:

Let the four parts be denoted as \begin{gathered}x_1, x_2, x_3\\\end{gathered}

x

1

,x

2

,x

3

and \begin{gathered}x_4\\\end{gathered}

x

4

.

As per the given data, \begin{gathered}4times\\\end{gathered}

4times

the first part is equal to \begin{gathered}12 times\\\end{gathered}

12times

the fourth part. This can be represented as follows:

⇒\begin{gathered}4x_1=12x_4\\\end{gathered}

4x

1

=12x

4

⇒\begin{gathered}x_1=3x_4\\\end{gathered}

x

1

=3x

4

.......(1)

\begin{gathered}3times\\\end{gathered}

3times

the second part is equal to \begin{gathered}12 times\\\end{gathered}

12times

the fourth part. This is represented as follows:

\begin{gathered}3x_2=12x_4\\x_2=4x_4\\\end{gathered}

3x

2

=12x

4

x

2

=4x

4

........(2)

Also, given that, \begin{gathered}2times\\\end{gathered}

2times

the third part is equal to \begin{gathered}12 times\\\end{gathered}

12times

the fourth part.

⇒\begin{gathered}2x_3=12x_4\\\end{gathered}

2x

3

=12x

4

⇒\begin{gathered}x_3=6x_4\\\end{gathered}

x

3

=6x

4

........(3)

The first three parts are written in terms of the fourth part which are represented in equations (1), (2) and (3).

Now, from the given data, the sum of four parts is equal to \begin{gathered}784\\\end{gathered}

784

. This means:

\begin{gathered}x_1+x_2+x_3+x_4=784\\\end{gathered}

x

1

+x

2

+x

3

+x

4

=784

........(4)

Substitute the equations (1), (2) and (3) in the equation (4). We get:

\begin{gathered}3x_4+4x_4+6x_4+x_4=784\\14x_4=784\\x_4=56\\\end{gathered}

3x

4

+4x

4

+6x

4

+x

4

=784

14x

4

=784

x

4

=56

⇒The value of \begin{gathered}x_4\\\end{gathered}

x

4

is \begin{gathered}56\\\end{gathered}

56

.

Now, substitute the value of \begin{gathered}x_4\\\end{gathered}

x

4

in the equation (1).

\begin{gathered}x_1=3(x_4)\\x_1=3(56)\\x_1=168\\\end{gathered}

x

1

=3(x

4

)

x

1

=3(56)

x

1

=168

Final Answer:

The required value of the first part is \begin{gathered}168\\\end{gathered}

168

.