## If \$a\$, \$b\$, \$c\$, and \$d\$ are replaced by four distinct digits from \$1\$ to \$9\$, inclusive, then what’s the largest possible value of the dif

Question

If \$a\$, \$b\$, \$c\$, and \$d\$ are replaced by four distinct digits from \$1\$ to \$9\$, inclusive, then what’s the largest possible value of the difference \$a.b – c.d\$ ?

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18 mins 2021-09-12T09:53:09+00:00 2 Answers 0

70

Step-by-step explanation:

• for the value of difference to be largest, the minuend should be maximum(most possibly) and the subtrahend should be minimum

[in A-B=X, A is minuend and B is subtrahend ]

• so, \$a.b should be maximum. as there is a condition that 4 digits should be distinct, the product will be maximum if we choose 2 maximum valued numbers from the given numbers. so, one of them should be 9 and the other should be 8.

therefore, \$a.b=9*8=72

• as mentioned above, c.d\$ should be minimum. this will be possible only when we choose  2 minimum valued numbers from the given numbers. so, one of them should be 1 and the other should be 2.

therefore, c.d\$ = 1*2 = 2

• hence, the difference = 72-2 = 70
• thus,  the largest possible value of the difference \$a.b – c.d\$ = 70

18.3

Step-by-step explanation:

Given that a, b, c, and d are different numbers (from 1 to 9), we are to find the largest possible value of a.b + c.d.

This problem can be answered by trial and error and with some logic. Clearly, a to d cannot be at the lower end of the values (1 to 5). Since the digits must be different, it can be a combination of digits from 6 to 9.

It is rather tempting to say that the 9.8 + 7.6 would yield the highest possible sum (=17.4) but this is incorrect. Since it is the sum of two numbers with a decimal, we have to maximize on the one-digit first before maximizing the tenths digit. Therefore: the combination of numbers must then be:

9.7 + 8.6 which results to 18.3

9.6 + 8.7 yields 18.3 as well.