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If two 2-digit positive integers have their respective tens digits exc

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Re: If two 2-digit positive integers have their respective tens digits exc  [#permalink]

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New post 22 Oct 2019, 19:36
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dcummins wrote:

I was a bit confused with the last step here.

What are the constraints on the numbers we must choose to determine that the original difference would be 78?

I also solved as (10a + b) - (10c +d) = 10c+b - (10a+d) + 4 to produce (10a -10c) = 2


According to our calculations, as long as the difference between the units digits is 2, the condition of "the difference between the original difference and the difference of the exchanged numbers is 4" is satisfied. That's the constraint of the units digit. For instance, 75 and 43 also satisfy that condition, but those numbers will not give you the greatest possible difference. In order to find "the greatest possible difference", the tens digit of the greater number must be 9 and the tens digit of the smaller number must be 1. As long as these two consitions are satisfied, you can choose any numbers you want (such as 92 and 10, 93 and 11, 94 and 12 etc.) The original difference is always 82 and the new difference is always 78.

In your equation, if 10a + b is the greater number, then after the switch is made, 10a + d will be the greater number; hence the right hand side of your equation should be (10a + d) - (10c + b) + 4.
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Re: If two 2-digit positive integers have their respective tens digits exc  [#permalink]

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New post 22 Oct 2019, 21:19
ScottTargetTestPrep wrote:
dcummins wrote:

I was a bit confused with the last step here.

What are the constraints on the numbers we must choose to determine that the original difference would be 78?

I also solved as (10a + b) - (10c +d) = 10c+b - (10a+d) + 4 to produce (10a -10c) = 2


According to our calculations, as long as the difference between the units digits is 2, the condition of "the difference between the original difference and the difference of the exchanged numbers is 4" is satisfied. That's the constraint of the units digit. For instance, 75 and 43 also satisfy that condition, but those numbers will not give you the greatest possible difference. In order to find "the greatest possible difference", the tens digit of the greater number must be 9 and the tens digit of the smaller number must be 1. As long as these two consitions are satisfied, you can choose any numbers you want (such as 92 and 10, 93 and 11, 94 and 12 etc.) The original difference is always 82 and the new difference is always 78.

In your equation, if 10a + b is the greater number, then after the switch is made, 10a + d will be the greater number; hence the right hand side of your equation should be (10a + d) - (10c + b) + 4.

+

thanks scott!
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Re: If two 2-digit positive integers have their respective tens digits exc  [#permalink]

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New post 24 Oct 2019, 19:15
dcummins wrote:
ScottTargetTestPrep wrote:
dcummins wrote:

I was a bit confused with the last step here.

What are the constraints on the numbers we must choose to determine that the original difference would be 78?

I also solved as (10a + b) - (10c +d) = 10c+b - (10a+d) + 4 to produce (10a -10c) = 2


According to our calculations, as long as the difference between the units digits is 2, the condition of "the difference between the original difference and the difference of the exchanged numbers is 4" is satisfied. That's the constraint of the units digit. For instance, 75 and 43 also satisfy that condition, but those numbers will not give you the greatest possible difference. In order to find "the greatest possible difference", the tens digit of the greater number must be 9 and the tens digit of the smaller number must be 1. As long as these two consitions are satisfied, you can choose any numbers you want (such as 92 and 10, 93 and 11, 94 and 12 etc.) The original difference is always 82 and the new difference is always 78.

In your equation, if 10a + b is the greater number, then after the switch is made, 10a + d will be the greater number; hence the right hand side of your equation should be (10a + d) - (10c + b) + 4.

+

thanks scott!


Sure thing!
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Re: If two 2-digit positive integers have their respective tens digits exc  [#permalink]

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New post 09 Dec 2019, 17:28
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Hi All,

We're told that if two 2-digit positive integers have their respective TENS digits exchanged, the difference between the pair of integers changes by 4. We're asked for the GREATEST possible difference between the original pair of integers. There are a few different ways to approach this question (and some of them are incredibly "step heavy"). Thankfully, we can solve it rather easily by TESTing THE ANSWERS.

Since we're looking for the GREATEST possible difference between the two 2-digit integers, we should start with Answer E. However, we can use some Number Properties to quickly eliminate a couple of the possibilities first.

The smallest 2-digit number is 10 and the largest is 99, so the difference between those two numbers is 89 (and can only get smaller). Thus, Answers D and E are NOT possible, so we do not have to consider them. Let's TEST Answer C first....

Answer C: 82
Let's pick two 2-digit numbers that have a difference of 82... the easiest would be 10 and 92.
We already know that the difference between these numbers is 82.
When we switch the TENS digits, we have 90 and 12. The difference between these numbers is 90 - 12 = 78.
The two results (82 and 78) differ by 4 - and this is an exact match for what we were told, so this MUST be the answer.

Final Answer:

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Re: If two 2-digit positive integers have their respective tens digits exc   [#permalink] 09 Dec 2019, 17:28

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