If \$x\$ and \$y\$ are the 10s and the systems digit, respectively, of the product \$725,278×67,066\$, what is the value of \$x+y\$ ?\$12\$\$10\$\$8\$\$6\$\$4\$

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## Survey the Question

Let’s search the problem for clues as to what it will certainly be experimentation, as this will help change our minds to think around what kind of math expertise we’ll usage to solve this question. Pay attention to any type of words that sound math-particular and also anypoint distinct around what the numbers look choose, and also mark them on our paper.

You are watching: If x and y are the tens digit and the units digit respectively

We desire to multiply two numbers together and also look at what numbers are in the devices and tens places, which will certainly involve things we’ve learned about Integers.

## What Do We Know?

Let’s very closely check out through the question and make a list of the points that we know.

We will multiply a \$6\$-digit number by a \$5\$-digit numberWe want to know that product’s sum of its devices and also tens digits

## Develop a Plan

Let’s assault this difficulty through a top-dvery own strategy. We want to understand the sum of the units and also 10s digits of the product of the numbers \$725,278·67,066\$. There is one solution easily easily accessible to us: we might simply multiply the numbers together to obtain the systems and tens digits of the product. That does seem prefer a time-consuming process, particularly if our calculator can’t take care of such a huge product (prefer, say, the measly GRE calculator offered to us on test day). So let’s try to discover an additional more clever before strategy to resolving this question.

We might remember from when we first learned lengthy multiplication that the units digit of a product is only established by the systems digits of the numbers being multiplied. Also, the tens digit of a product is figured out by the devices and also 10s digits of the numbers being multiplied. If we’re curious around why this is, then let’s feel free to look at it in even more depth. Otherwise we have the right to simply go ahead and settle this question.

## Concept Refresher – Units and also Tens Digits for Multiplication

Due to the fact that we just treatment around the units and also tens digits, maybe we would be much better off looking at numbers particularly based upon their digits, so let’s separation one of the numbers up by its digits. We recognize that each digit represents a factor of \$10\$, so let’s work on separating up one of the numbers by writing it as the amount of components of \$10\$. It doesn’t matter which number we perform this for, so we’ll arbitrarily pick \$725,278\$. So let’s first compose it as the sum of components of \$10\$:

\$\$725,278 = 8 + 70 + 200 + 5,000 + 20,000 + 700,000\$\$

Now let’s create out the product that we’re looking for making use of this:

\$\$725,278·67,066 = 67,066·(8 + 70 + 200 + 5,000 + 20,000 + 700,000)\$\$

And we know that we have the right to use circulation to recompose this as the sum of multiple assets, by distributing the \$67,066\$ inside the parentheses.

\$\$725,278·67,066 = 67,066·8 + 67,066·70 + 67,066·200 + …\$\$

We understand that when we multiply two numbers together, any type of \$0\$’s at the finish of the numbers will be tbelow in the product. So for example, \$5·700 = 5·7·100 = 3,500\$. So for the 3 commodities that involve \$200\$, \$5,000\$, \$20,000\$ and \$700,000\$, we know that they will certainly all have actually \$0\$’s for their systems and tens digits. This would certainly also hold true for later as soon as we multiply by \$67,066\$, as we could additionally split this number up by its digits, focusing on the multiplication via the devices and 10s digits. So they will include nopoint to the sum of the devices and also tens digits for \$725,278·67,066\$. For these reasons, we’ll break-up both numbers up keeping the systems and also 10s digits sepaprice from the rest of the number, then multiply them together. Now let’s gain earlier to the question at hand.

## Solve the Question

So we know that we just need to usage the 10s and units digits in this multiplication to obtain the tens and also units digits of the product, so let’s choose the tens and also units digits of the two numbers and multiply them together.

\$\$78·66 = 5,148\$\$

So the units and also 10s digits are \$4\$ and \$8\$. Due to the fact that \$4+8=12\$, the sum of the devices and also tens digits is \$12\$, and also the correct answer is A, \$12\$.

## What Did We Learn

An vital lesboy to learn from this trouble is that we did NOT have to perform the entire product simply to discover the amount of the systems and tens digits. So currently we know that for future products wbelow we just desire to know the systems and tens digits, we deserve to simply overlook anypoint beyond the 10s digit.

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