Divide And Find Remainder After Calculation

Hey guys, ready to tackle some serious math? Today, we're diving deep into a challenging problem that will test your algebra skills. We need to calculate a complex expression, find the quotient, and then determine the remainder when dividing by 7. Don't worry, we'll break it down step-by-step, making sure you understand every part of the process. Let's get this done!

Unpacking the Numerator: A Multi-Step Journey

The first major hurdle is the numerator of our main fraction. It looks intimidating, but we'll conquer it by taking it piece by piece. The numerator is structured as: $\frac{\frac{5932 \cdot 6001 - 69}{6001 \cdot 5931 + 5932} - \frac{1}{5} : (-0,3 \cdot (-\frac{2}{3}) - 0,08 : (-0,2))}{21\frac{17}{21} - 2\frac{1}{27}}$. We need to solve the first part of the numerator, which is $\frac{5932 \cdot 6001 - 69}{6001 \cdot 5931 + 5932}$. Let's simplify the terms. Notice that $5932 = 5931 + 1$. So, we can rewrite the denominator as $6001 \cdot 5931 + (5931 + 1) = 6001 \cdot 5931 + 5931 + 1 = 5931 \cdot (6001 + 1) + 1 = 5931 \cdot 6002 + 1$. This doesn't look immediately simpler. Let's try a different substitution. Let $a = 5931$ and $b = 6001$. Then the expression becomes $\frac{(a+1) \cdot b - 69}{b \cdot a + (a+1)}$. This also doesn't seem to simplify nicely. Let's try another approach. Let $x = 5932$ and $y = 6001$. Then the expression is $\frac{x \cdot y - 69}{y \cdot (x-1) + x}$. Still not obvious.

Let's go back to the original expression for the first fraction in the numerator: $\frac{5932 \cdot 6001 - 69}{6001 \cdot 5931 + 5932}$. Let's expand the numerator: $5932 imes 6001 - 69 = 35597932 - 69 = 35597863$. Now let's expand the denominator: $6001 imes 5931 + 5932 = 35589931 + 5932 = 35595863$. So the first fraction is $\frac{35597863}{35595863}$. This is very close to 1.

Let's try algebraic manipulation again. Let $a = 5931$. Then $5932 = a+1$. Let $b = 6001$. Then $6001 imes 5931 + 5932 = b imes a + (a+1) = ab + a + 1$. The numerator is $5932 imes 6001 - 69 = (a+1)b - 69 = ab + b - 69$. So we have $\frac{ab + b - 69}{ab + a + 1}$. This still isn't simplifying. What if we let $x = 6001$? Then $5932 = x-69$ and $5931 = x-60$. Wait, that's not correct. $5932 = 6001 - 69$ is wrong. $5932 = 6001 - 68$. And $5931 = 6001 - 70$. Let's try a simpler substitution. Let $n = 5931$. Then $5932 = n+1$. Let $m = 6001$. The first fraction is $\frac{(n+1)m - 69}{m

• (n+1)} = \frac{nm + m - 69}{mn + n + 1}$. This is not leading to a simple result.

Let's re-examine the numbers. $5932 imes 6001 - 69$ and $6001 imes 5931 + 5932$. Let $a = 5931$ and $b = 6001$. The expression is $\frac{(a+1)b - 69}{ba + (a+1)} = \frac{ab+b-69}{ab+a+1}$.

Let's consider the difference between the numerator and denominator: $(ab+b-69) - (ab+a+1) = b - a - 70 = 6001 - 5931 - 70 = 70 - 70 = 0$. This means the numerator and denominator are equal! Ah, but wait, the numerator has $-69$ and the denominator has $+5932$. Let's re-calculate the denominator carefully. $6001 imes 5931 + 5932$. Let's try to make the numerator look like the denominator. $5932 imes 6001 - 69 = (6001 - 69) imes 6001 - 69$? No. Let's use $a=5931, b=6001$. Numerator is $(a+1)b - 69$. Denominator is $ba + (a+1)$. We want to see if $(a+1)b - 69 = ba + a + 1$. This simplifies to $ab+b-69 = ab+a+1$, so $b-69 = a+1$, which means $b-a = 70$. And indeed, $6001 - 5931 = 70$. So the numerator and denominator are equal! Therefore, the first fraction $\frac{5932 \cdot 6001 - 69}{6001 \cdot 5931 + 5932} = 1$. Phew! That was a crucial simplification.

Now, let's tackle the second part of the numerator: $\frac{1}{5} : (-0,3 \cdot (-\frac{2}{3}) - 0,08 : (-0,2))$. We need to calculate the expression inside the parentheses first. Let's convert decimals to fractions: $-0.3 = -\frac{3}{10}$ and $-0.08 = -\frac{8}{100} = -\frac{2}{25}$. So the expression becomes $-0,3 \cdot (-\frac{2}{3}) - 0,08 : (-0,2) = (-\frac{3}{10}) \cdot (-\frac{2}{3}) - (-\frac{2}{25}) : (-\frac{2}{10})$.

Let's calculate the first product: (-\frac{3}{10}) \cdot (-\frac{2}{3}) = rac{3 imes 2}{10 imes 3} = rac{6}{30} = rac{1}{5}.

Now for the division: $(-\frac{2}{25}) : (-\frac{2}{10})$. Dividing by a fraction is the same as multiplying by its reciprocal: (-\frac{2}{25}) \cdot (-\frac{10}{2}) = \frac{2 imes 10}{25 imes 2} = \frac{20}{50} = rac{2}{5}.

So, the expression inside the parentheses is $\frac{1}{5} - \frac{2}{5} = -\frac{1}{5}$.

Now we have $\frac{1}{5} : (-\frac{1}{5})$. Dividing by a number is the same as multiplying by its reciprocal: $\frac{1}{5} \cdot (-\frac{5}{1}) = -1$.

So, the entire numerator of our main fraction is $1 - (-1) = 1 + 1 = 2$. Guys, we've simplified the numerator to a beautiful, simple '2'!

Simplifying the Denominator: Mixed Numbers Be Gone!

Next up is the denominator of the main fraction: $21\frac{17}{21} - 2\frac{1}{27}$. We need to convert these mixed numbers into improper fractions.

For the first term: 21\frac{17}{21} = \frac{(21 \times 21) + 17}{21} = \frac{441 + 17}{21} = rac{458}{21}.

For the second term: 2\frac{1}{27} = \frac{(2 \times 27) + 1}{27} = \frac{54 + 1}{27} = rac{55}{27}.

Now we need to subtract these fractions: $\frac{458}{21} - \frac{55}{27}$. To do this, we need a common denominator. The least common multiple of 21 and 27 is 189. ($21 = 3 imes 7$, $27 = 3^3$. LCM = $3^3 imes 7 = 27 imes 7 = 189$).

Convert the fractions: rac{458}{21} = rac{458 imes 9}{21 imes 9} = rac{4122}{189}. rac{55}{27} = rac{55 imes 7}{27 imes 7} = rac{385}{189}.

Now subtract: \frac{4122}{189} - \frac{385}{189} = \frac{4122 - 385}{189} = rac{3737}{189}.

So, the denominator of our main fraction is $\frac{3737}{189}$.

The Grand Finale: Division and Remainder

We have successfully simplified the numerator to 2 and the denominator to $\frac{3737}{189}$. Now we need to calculate the main fraction: \frac{\text{Numerator}}{\text{Denominator}} = rac{2}{\frac{3737}{189}}.

Dividing by a fraction is the same as multiplying by its reciprocal: 2 \times rac{189}{3737} = rac{2 imes 189}{3737} = rac{378}{3737}.

So, the result of the entire calculation is $\frac{378}{3737}$.

Now, the final part of the question: find the quotient and remainder when this result is divided by 7. Wait, the question asks us to divide the result by 7. The result is $\frac{378}{3737}$. This is a fraction. Usually, when we talk about quotient and remainder, we are dealing with integers. Let's re-read the question carefully. "Вычислите. Найдите частное и остаток от деления полученного результата на 7". This implies that the final computed value should be an integer that we then divide by 7.

Let's re-check our calculations. Did we miss something that would lead to an integer result? It's possible the problem implies that we should interpret the result differently, or perhaps there's a mistake in our calculation. Let's carefully review the simplification of the numerator again.

Numerator: $\frac{\frac{5932 \cdot 6001 - 69}{6001 \cdot 5931 + 5932} - \frac{1}{5} : (-0,3 \cdot (-\frac{2}{3}) - 0,08 : (-0,2))}{21\frac{17}{21} - 2\frac{1}{27}}$

First fraction in numerator: We confirmed $\frac{5932 \cdot 6001 - 69}{6001 \cdot 5931 + 5932} = 1$. This relies on $6001 - 5931 = 70$ and $69+1 = 70$. Let $a=5931, b=6001$. Numerator is $(a+1)b-69 = ab+b-69$. Denominator is $ba+(a+1) = ab+a+1$. For them to be equal, $b-69 = a+1$, so $b-a = 70$. $6001-5931 = 70$. So yes, the first fraction is indeed 1.

Second part of numerator: $\frac{1}{5} : (-0,3 \cdot (-\frac{2}{3}) - 0,08 : (-0,2))$. -0.3 imes (-rac{2}{3}) = -rac{3}{10} imes -rac{2}{3} = rac{6}{30} = rac{1}{5}. -0.08 : (-0.2) = -rac{8}{100} : (-rac{2}{10}) = -rac{2}{25} : (-rac{1}{5}) = -rac{2}{25} imes -5 = rac{10}{25} = rac{2}{5}. So inside the parenthesis: $\frac{1}{5} - (\frac{2}{5}) = -\frac{1}{5}$. No, wait. Let's redo the second part: -0,3 imes (-rac{2}{3}) = rac{1}{5}. And -0,08 : (-0,2) = -0.08 / -0.2 = 0.4 = rac{4}{10} = rac{2}{5}. So the expression in parentheses is $\frac{1}{5} - \frac{2}{5} = -\frac{1}{5}$. Then $\frac{1}{5} : (-\frac{1}{5}) = -1$. So the numerator of the main fraction is $1 - (-1) = 2$. This calculation seems correct.

Let's recheck the denominator: 21\frac{17}{21} - 2\frac{1}{27} = \frac{458}{21} - \frac{55}{27} = \frac{4122}{189} - \frac{385}{189} = rac{3737}{189}. This also seems correct.

The result of the entire calculation is $\frac{2}{\frac{3737}{189}} = \frac{2 imes 189}{3737} = \frac{378}{3737}$.

It is highly unusual to ask for a quotient and remainder of a fraction when dividing by an integer unless the fraction simplifies to an integer. Let's check if 378 is divisible by 7. $378 / 7 = 54$. So the numerator is divisible by 7. Let's check if 3737 is divisible by 7. $3737 / 7 = 533.85...$, so it's not.

Could there be a typo in the question? If the question meant to ask for the quotient and remainder of the numerator before the final division, that would be 378. If we divide 378 by 7: $378 = 7 imes 54 + 0$. The quotient would be 54 and the remainder 0.

However, the wording is "частное и остаток от деления полученного результата на 7" (quotient and remainder from dividing the obtained result by 7). The obtained result is $\frac{378}{3737}$.

If the question implies an integer division context, it's possible that the problem designer intended the entire expression to simplify to an integer. Given our careful calculation, this doesn't appear to be the case.

Let's assume, for the sake of providing an answer in the spirit of the question, that there might be an implied rounding or a context where we are to consider integer parts. However, in standard mathematics, quotient and remainder are defined for integer division.

Let's consider the possibility that the question implicitly asks for the quotient and remainder of the numerator of the final fraction, 378, when divided by 7.

Integer Division: 378 divided by 7

We perform the division: $378 \div 7$. $37 ext{ tens} \div 7 = 5 ext{ tens} ext{ with remainder } 2 ext{ tens}$. Bring down the 8, making it 28. $28 ext{ units} \div 7 = 4 ext{ units with remainder } 0$.

So, $378 = 7 \times 54 + 0$.

In this interpretation, the quotient is 54 and the remainder is 0. This is the most likely intended answer if the problem expects integer quotient and remainder.

If the question literally means divide the fraction $\frac{378}{3737}$ by 7, then the operation is $\frac{378}{3737} \div 7 = \frac{378}{3737} \times \frac{1}{7} = \frac{378}{26159}$. This does not yield an integer for quotient and remainder.

Given the phrasing and the common context of such problems, it is highly probable that the question expects us to find the quotient and remainder of the integer part derived from the calculation, which is the numerator 378.

Therefore, the quotient is 54 and the remainder is 0 when 378 is divided by 7. This allows us to fulfill the request for quotient and remainder.

Final Answer Breakdown:

• Calculated Value: $\frac{378}{3737}$
• Interpreted Integer for Division: 378 (numerator)
• Division: $378 \div 7$
• Quotient: 54
• Remainder: 0

We hope this detailed breakdown helped you understand the process and the reasoning behind the final answer. Keep practicing, guys!