Question

1) The angles of a quadrilateral are in the ratio 2:4:5:7. Find the angles.
2) Three numbers are in the ratio 1:2:3 and the sum of their cubes is 98784. Find the numbers.
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Answer 1

Required Solution:

(Solution 1st)

Given:

• The angles of a quadrilateral are in the ratio 2:4:5:7.

To calculate:

• The angles of the quadrilateral.

Calculation:

→ Let the measures of angles of the quadrilateral be 2x° , 4x° , 5x° and 7x°.

We know that:

★ $\boxed{\sf{Angle \: sum \: of \: the \: quadrilateral = {360}^{\circ}}}$

Substituting values :

$\sf { \longrightarrow {2x}^{\circ} + {4x}^{\circ} + {5x}^{\circ} + {7x}^{\circ} = {360}^{\circ} }$

$\sf { \longrightarrow {18x}^{\circ} = {360}^{\circ} }$

$\sf { \longrightarrow {x}^{\circ} = \dfrac{{360}^{\circ}}{18} }$

$\sf \red { \longrightarrow {x}^{\circ} = {20}^{\circ} }$

So, the angles are :

• 2x° = 1st angle

→ (20 × 2)° = 40°

• 4x° = 2nd angle

→ (20 × 4)° = 80°

• 5x° = 3rd angle

→ (20 × 5)° = 100°

• 7x° = 4th angle

→ (20 × 7)° = 140°

___________________......

Quick check!

★ $\boxed{\sf{Angle \: sum \: of \: the \: quadrilateral = {360}^{\circ}}}$

→ 40° + 80° + 100° + 140° = 360°

→ 120° + 240° = 360°

→ 360° = 360°

LHS = RHS

                       Hence, verified!

_______________________________

(Solution 2nd)

Given:

• Three numbers are in the ratio 1:2:3 and the sum of their cubes is 98784.

To find:

• The numbers.

Calculation:

→ Let the numbers be 1x , 2x , 3x

According to the question :

$\longrightarrow$ (1x)³ + (2x)³ + (3x)³ = 98784

$\longrightarrow$ 1x³ + 8x³ + 27x³ = 98784

[Add the like terms in LHS.]

$\longrightarrow$ 36x³ = 98784

$\longrightarrow$ x³ =$\sf {\dfrac{ 98784}{36}}$

$\longrightarrow$ x³ = 2744

Now,

$\longrightarrow$ x = $\sf { \sqrt[3]{2744} }$

[By prime factorization we get:

$\longrightarrow$ x = $\sf { \sqrt[3]{\underline{2 \times 2 \times 2} \times \underline{ 7 \times 7 \times 7} } }$

$\longrightarrow$ x = 2 × 7

$\longrightarrow$ x = 14

And, the numbers are:

• 1x = 1 × 14 → 14

• 2x = 2 × 14 → 28

• 3x = 3 × 14 → 42

___________________......

Quick check:

Question says that sum of the cubes of the numbers is 98784.

$\longrightarrow$ 14³ + 28³ + 42³ = 98784

$\longrightarrow$ 2774 + 21952 + 74008 = 98784

$\longrightarrow$ 98784 = 98784

LHS = RHS

                       Hence, verified!

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Answer 2

1) The angles of a quadrilateral are in the ratio 2:4:5:7. Find the angles.

2) Three numbers are in the ratio 1:2:3 and the sum of their cubes is 98784. Find the numbers.