Question

Six years ago, the age of mother was equal to the square of her son's age. after Three yeare,
her age will be thrice the age of her son then. Find the present ages of mother and son ​

Answer 1

$\LARGE\underline\mathfrak{AnSwEr:}$

• ᴘʀᴇsᴇɴᴛ ᴀɢᴇ ᴏғ sᴏɴ = 12 ʏʀs

• ᴘʀᴇsᴇɴᴛ ᴀɢᴇ ᴏғ ᴍᴏᴛʜᴇʀ = 42 ʏʀs

$\LARGE\underline\mathfrak{Step \: by \: step \: explanation :}$

ʟᴇᴛ ᴛʜᴇ ᴘʀᴇsᴇɴᴛ ᴀɢᴇ ᴏғ sᴏɴ ʙᴇ $\small\tt\ x \: yrs$

So,

ᴀɢᴇ ᴏғ ᴍᴏᴛʜᴇʀ ᴡɪʟʟ ʙᴇ $\small\tt\ x^{2} \: yrs$

Now,

ᴘʀᴇsᴇɴᴛ ᴀɢᴇs ɴᴏᴡ ;( six years ago)

➠ᴀɢᴇ ᴏғ sᴏɴ = $\small\tt\ x \: + \: 6 \: yrs$

➠ᴀɢᴇ ᴏғ ᴍᴏᴛʜᴇʀ = $\small\tt\ x^{2} \: + \: 6 \: yrs$

3 ʏᴇᴀʀs ʟᴀᴛᴇʀ ; ( after more three years)

➠ᴀɢᴇ ᴏғ sᴏɴ = $\small\tt\ x \: + 6 \: + \: 3 \: = \: x \: + 9 \: yrs$

➠ᴀɢᴇ ᴏғ ᴍᴏᴛʜᴇʀ = $\small\tt\ x^{2} \: + \: 6 \: + \: 3 \: = \: x^{2} \: + \: 9 \: yrs$

ᴀᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ǫᴜᴇsᴛɪᴏɴ ;

• ᴀɢᴇ ᴏғ ᴍᴏᴛʜᴇʀ ɪs 3 ᴛɪᴍᴇs ᴏʀ ᴛʜʀɪᴄᴇ ᴛʜᴇ ᴀɢᴇ ᴏғ sᴏɴ

➠$\large\tt\ x^2 \: + \: 9 \: = \: 3(x \: + \: 9)$

➠$\large\tt\ x^2 \: + \: 9 \: = \: 3x \: + 27$

➠$\large\tt\ x^2 \: - \: 3x \: - \: 18 \: = 0$

• ᴜsɪɴɢ ᴍɪᴅᴅʟᴇ ᴛᴇʀᴍ ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ

➠$\large\tt\ x^2 \: -6x \: + \: 3x \: - \: 18 \: = \: 0$

➠$\large\tt\ x(x \: - \: 6) \: + \: 3(x \: - \: 6) \: = \: 0$

➠$\large\tt\ (x \: - \: 6)(x \: + \: 3) \: = \: 0$

➠$\large\tt\ x \: = \: 6, -3$

• -3 ɪs ᴛʜᴇ ʀᴇᴊᴇᴄᴛᴇᴅ ᴠᴀʟᴜᴇ,ᴡᴇ ᴄᴀɴ'ᴛ ᴜsᴇ ɪᴛ ᴛᴏ ғɪɴᴅ ᴀɢᴇs ( ᴀɢᴇs ᴀʀᴇ ɴᴏᴛ ɴᴇɢᴀᴛɪᴠᴇ),sᴏ ᴡᴇ ғɪɴᴅ ᴛʜᴇ ᴀɢᴇs ʙʏ ᴜsɪɴɢ 6.

➠ ᴀɢᴇ ᴏғ sᴏɴ :

$\rightarrow\tt\ x \: + \: 6 \: = \: 6 \: + \: 6 \: = \: 12 \: yrs$

➠ ᴀɢᴇ ᴏғ ᴍᴏᴛʜᴇʀ :

$\rightarrow\tt\ x^2 \: + \: 6 \: = \: 6^2 \: + \: 6 \: = \: 42 \: yrs$