Question

20ഓറഞ്ച് 20 ആൾക്കാരും ഉണ്ട്. കുട്ടികൾക്ക് 1/4 കൊടുത്തു.adult ന് നേർപകുതി. വയസ് കൂടിയവർക്ക് 4 എണ്ണം.അങ്ങനെ എത്ര ആൾക്കാർ ഉണ്ട് കണ്ടുപിടിക്കണം

Answer 1

Question in English:-

There are 20 oranges and 20 people. Children are given (1/4)th of oranges, adults are given half of them and elders are given 4 oranges. Determine the no. of children, adults and elders.

Solution:-

Let x, y and z be no. of children, adults and elders respectively.

Since there are 20 people,

$\small\longrightarrow x+y+z=20$

Given that there are 20 oranges, and children are given (1/4)th of oranges, adults are given half of them and elders are given 4 oranges. So,

$\small\longrightarrow\dfrac{x}{4}+\dfrac{y}{2}+4z=20$

Multiplying both sides by 4,

$\small\longrightarrow x+2y+16z=80$

Now we have two plane equations. We need to find the equation of intersection line of these two planes, then the positive integer solution(s) (x, y, z) of that line.

The equations of the planes are,

• $\smallP_1:\vec{r}\cdot\left<1,\ 1,\ 1\right>=20$

• $\smallP_2:\vec{r}\cdot\left<1,\ 2,\ 16\right>=80$

We see the normal vectors are $\small\left<1,\ 1,\ 1\right>$ and $\small\left<1,\ 2,\ 16\right>.$

The intersection line is perpendicular to both the normal vectors, or in other words it is parallel to cross product of the normal vectors, i.e.,

$\small\longrightarrow\vec{b}=\left|\begin{array}{ccc}\hat i&\hat j&\hat k\\1&1&1\\1&2&16\end{array}\right|$

$\small\longrightarrow\vec{b}=\left<14,\ -15,\ 1\right>$

Now we need to find a point on the line. We assume z = 0 such that we're finding the point at which our intersection line cuts XY plane.

Then plane equations are reduced to,

$\small\longrightarrow x+y=20$

$\small\longrightarrow x+2y=80$

Solving them we get,

• $\smallx=-40$

• $\smally=60$

Hence the intersection line is in the vector form,

$\small\longrightarrow\vec{r}=\left<-40,\ 60,\ 0\right>+\lambda\left<14,\ -15,\ 1\right>$

$\small\longrightarrow\vec{r}=\left<-40+14\lambda,\ 60-15\lambda,\ \lambda\right>$

Now we find integer value of λ such that each coordinate should be 'positive' integer.

As x coordinate is positive,

$\small\longrightarrow-40+14\lambda>0$

$\small\longrightarrow\lambda>\dfrac{40}{14}$

$\small\Longrightarrow\lambda\geq3\quad\dots(1)$

As y coordinate is positive,

$\small\longrightarrow60-15\lambda>0$

$\small\longrightarrow\lambda<4$

$\small\Longrightarrow\lambda\leq3\quad\dots(2)$

As z coordinate is positive,

$\small\longrightarrow\lambda>0$

$\small\Longrightarrow\lambda\geq1\quad\dots(3)$

Taking (1) ∧ (2) ∧ (3),

$\small\longrightarrow\lambda=3$

$\small\Longrightarrow\vec{r}=\left<-40+14(3),\ 60-15(3),\ 3\right>$

$\small\text{ \longrightarrow\underline{\underline{\vec{r}=\left<2,\ 15,\ 3\right>}} }$

Hence there are 2 children, 15 adults and 3 elders.

Answer 2

Answer:

There are 2 children. 15 adults and ig 2 elders.

HOPE IT HELPS!!

BE BRAINLY