Question

When
a × b=2 ,
b × c =3 ,
c × d=5 ,
d × e=6 ,
find e/a​
​

Answer 1

Key

Isolation by multiplication and division.

• Sol. 1: Isolation
• Sol. 2: System equation

Solution

Sol. 1

Multiplying all,

$ab \times bc \times cd \times de =2 \times 3 \times 5 \times 6$

$\Leftrightarrow a \times b^2 \times c^2 \times d^2 \times e=180$ ...(I)

Multiplying $ab$ and $cd$,

$ab \times cd =2 \times 5$

$\Leftrightarrow a \times b \times c \times d = 10$ ...(II)

Then,

$\dfrac{e}{a} =\dfrac{ab^2c^2d^2e}{a^2b^2c^2d^2}$

$= \dfrac{ab^2c^2d^2e}{(abcd)^2}$

By (I) and (II),

$\dfrac{e}{a} = \dfrac{180}{100}$

$=\dfrac{9}{5}$

The value of $\dfrac{e}{a}$ is hence $\dfrac{9}{5}$.

Sol. 2

Suppose $e \times a = k$.

Then,

$\begin{cases}&ab= 2\\ &bc= 3\\ &cd= 5\\ &de= 6\\ &ea= k\end{cases}$

Multiplying all,

$(abcde)^2=180k$

$\Leftrightarrow abcde = \pm 6\sqrt{5k}$ ...(I)

Now we isolate each variable using products.

• $bc \times de=18$ (without $a$)
• $cd \times ea=5k$ (without $b$)
• $de \times ab=2k$ (without $c$)
• $bc \times ea = 3k$ (without $d$)
• $ab \times cd = 10$ (without $e$)

Divide (I) by each product to isolate the variables.

$\therefore \begin{cases}&a= \pm \dfrac{6\sqrt{5k} }{18} \\ &b= \pm \dfrac{6\sqrt{5k} }{5k} \\ &c= \pm \dfrac{6\sqrt{5k} }{2k} \\ &d= \pm \dfrac{6\sqrt{5k} }{3k} \\ &e= \pm\dfrac{6\sqrt{5k} }{10} \end{cases}$

Then,

$\dfrac{e}{a} =(\pm \dfrac{\cancel{6\sqrt{5k} }}{10} ) \times (\pm \dfrac{18}{\cancel{6\sqrt{5k} }} )$

$=\dfrac{18}{10}$

$=\dfrac{9}{5}$

Hence the value is $\dfrac{9}{5}$.

More information:

Isolation is used in system equation solving. For example,

$\begin{cases}&x+y= 3\\ &y+z= 3\\ &z+x= 3\\ \end{cases}$

Adding all we obtain two times the sum. Divide by 2 to remove the multiplier.

$x+y+z=\dfrac{9}{2}$

Then we subtract by each value, then the variables are isolated.

$\therefore x=y=z=\dfrac{3}{2}$

Answer 2

> e/a = 9/5 is the required answer.

✯ Correct Question ✯

• When
• a × b=2 ,
• b × c =3 ,
• c × d=5 ,
• d × e=6 ,
• find e/a

✯ Given ✯

• a x b = 2_______(1)
• b x c = 3_______(2)
• c x d = 5_______(3)
• d x e = 6_______(4)

✯ Approach ✯

• We will equate the above equation to find the required answer.

✯ To find ✯

• find e/a

✯ Formula ✯

• Not any formula just Using Multiplication and division

✯ Solution ✯

Multiplying eq(1) with eq(3):

ac x bd = 10________(5)

Multiplying eq(4) with eq(2):

bd x ec = 18________(6)

Now,

dividing eq(5) with eq(6):

a/e= 10/18 = 5/9_____(7)

Reciprocal eq(7):

Hence,

⚽ e/a = 9/5 is the required answer. ⚽

✷ More ✷

> How to Approach puzzle question

• Take a quick look at the question.
• Develop a general idea regarding the theme of the problem.
• Select the data that is giving you some concrete information out of total information given.
• Also, select the data which helps in ruling out certain possibilities.