CAT 2024 Slot 3 · Quant · Question 13

Call the three numbers a<b<c.

Their average is 28, so a+b+c=84.

After the change the numbers are a+7, b and c-10, and the order is stated to be unchanged.

The new sum is 84+7-10=81, so the new mean is \frac{81}{3}=27.

The new mean is 2 more than the middle number, and the middle number is still b.

So 27=b+2, giving b=25.

The new difference between largest and smallest is \left( c-10 \right)-\left( a+7 \right)=c-a-17.

Setting this equal to 64 gives c-a=81.

From the original sum, a+c=84-25=59.

Adding c-a=81 and a+c=59 gives 2c=140, so c=70.

Then a=59-70=-11, and a quick check confirms -11<25<70 and -4<25<60 after the change.

The largest number in the original set is 70.

Shortcut. The middle number never moves, so the new mean pins b down on its own. After that it is just the sum and difference of a and c — two numbers from their sum 59 and difference 81.