Calculation of the Given Expression

Given the equation \(a^{2}+b^{2}=5ab\), we are tasked with determining the value of the expression \(\left(\frac{a^{2}}{b^{2}}+\frac{b^{2}}{a^{2}}\right)\).

Step 1: Simplify the Given Relation

Starting with the given equation, we divide both sides by \(ab\) to simplify:

\[
\frac{a^2 + b^2}{ab} = 5
\]

This leads to:

\[
\frac{a}{b} + \frac{b}{a} = 5
\]

Step 2: Square Both Sides

To find the value of \(\left(\frac{a^{2}}{b^{2}}+\frac{b^{2}}{a^{2}}\right)\), we square both sides of the simplified equation:

\[
\left(\frac{a}{b} + \frac{b}{a}\right)^2 = 5^2
\]

This yields:

\[
\frac{a^2}{b^2} + 2\left(\frac{a}{b}\cdot\frac{b}{a}\right) + \frac{b^2}{a^2} = 25
\]

Given that \(\frac{a}{b}\cdot\frac{b}{a} = 1\), we simplify further:

\[
\frac{a^2}{b^2} + \frac{b^2}{a^2} + 2 = 25
\]

Step 3: Isolate the Target Expression

Subtracting 2 from both sides to isolate the expression gives us:

\[
\frac{a^2}{b^2} + \frac{b^2}{a^2} = 25 – 2 = 23
\]

Conclusion

Therefore, the value of the expression \(\left(\frac{a^{2}}{b^{2}}+\frac{b^{2}}{a^{2}}\right)\) is \(\textbf{23}\), making the correct answer:

(c) \(\textbf{23}\)

This solution methodically derives the value of the given expression by leveraging the initial condition and algebraic manipulation, leading to a clear and logical conclusion.

Calculation of Water Levels in Jars with Different Capacities

This problem presents a scenario where equal volumes of water are introduced into two jars of distinct capacities. Here’s a breakdown of the situation and the outcome when the water from one jar is transferred to the other.

Initial Setup

– Larger Jar: When filled with a certain volume of water, it reaches \(\frac{1}{4}\) of its full capacity.
– Smaller Jar: The same volume of water fills this jar to \(\frac{1}{3}\) of its capacity, indicating its smaller size compared to the larger jar.

Transfer Process and Outcome

Upon transferring the water from the smaller jar (which is \(\frac{1}{3}\) full) into the larger jar (\(\frac{1}{4}\) full), we aim to understand how the water level changes in the larger jar.

– Observation: Since the initial amounts of water in both jars are equal, transferring the water from the smaller jar to the larger one effectively doubles the amount of water in the larger jar.

– Mathematical Representation: The act of pouring water from the smaller jar doubles the water volume in the larger jar, leading to the equation \(2 \times \frac{1}{4} = \frac{1}{2}\).

Conclusion

Therefore, after the water from the smaller jar is poured into it, the larger jar becomes \(\frac{1}{2}\) full.

To find the three consecutive odd numbers whose sum is 1383, let’s denote the smallest of these numbers as \(n\), the next one as \(n + 2\), and the largest as \(n + 4\) (since odd numbers differ by 2).

The sum of these numbers is given as:
\[n + (n + 2) + (n + 4) = 1383\]

Simplifying, we get:
\[3n + 6 = 1383\]

Subtracting 6 from both sides:
\[3n = 1377\]

Dividing by 3:
\[n = 459\]

So, the three consecutive odd numbers are 459, 461, and 463. The largest number among them is \(463\).

Therefore, the correct answer is (a) 463.

Given the constraint that the sum of the number and its reversed form is 888 and focusing on the clue that the units digit in the sum scenario must add up to 8, we look directly at the options provided:

– (a) 339: The sum of 3 and 9 does not lead to an end digit of 8 in the sum.
– (b) 341: The sum of 1 and 4 does not lead to an end digit of 8 in the sum.
– (c) 378: The sum of 8 and 7 does not directly address the end digit sum condition.
– (d) 345: When 345 is added to its reverse 543, the unit digits 5 and 3 indeed add up to 8, meeting the immediate condition.

The specific insight about the unit’s digits adding up to 8 being satisfied only by option (d) “345” (since \(3+5=8\)) simplifies the approach significantly. However, to validate this option fully in the context of the entire problem:

– If the original number is 345 and its reverse is 543, their sum is indeed 888 (\(345 + 543 = 888\)), which satisfies one of the problem’s conditions.
– The second condition mentioned is that swapping the unit’s and ten’s digit of the original number results in a number that is 9 more than the original. Swapping the units and tens digit of 345 gives 354, which is indeed 9 more than 345 (\(354 – 345 = 9\)).

Therefore, considering both conditions and the insight provided about the sum leading to the last digits adding up to 8, the correct number is indeed option (d) **345**. This choice fulfills both specified conditions of the problem: the sum with its reversed form equals 888, and swapping the tens and units digits results in a number that is 9 more than the original.

Let’s denote the two numbers as \(x\) and \(y\). We are given that:

1. The sum of the two numbers is 42:
\[x + y = 42\]

2. Their product is 437:
\[xy = 437\]

We are asked to find their difference, which is \(|x – y|\).

To find \(x\) and \(y\), we can use the given sum and product. These can be considered as the sum and product of the roots of a quadratic equation, which can be written as:

\[x^2 – (x + y)x + xy = 0\]

Substituting the given values:

\[x^2 – 42x + 437 = 0\]

Now, we solve this quadratic equation for \(x\). The solutions to the equation will give us the values of \(x\) and \(y\). Instead of directly solving the quadratic equation, let’s use another approach to find the difference, as solving it might get a bit complex without a calculator.

The difference squared can be found using the formula derived from the sum and product of roots:
\[(x – y)^2 = (x + y)^2 – 4xy\]

Given \(x + y = 42\) and \(xy = 437\), we substitute these values into the equation:
\[(x – y)^2 = 42^2 – 4 \cdot 437\]

Calculating the right side:
\[(x – y)^2 = 1764 – 1748\]

\[(x – y)^2 = 16\]

Taking the square root of both sides:
\[|x – y| = 4\]

Therefore, the difference between the two numbers is \(\boldsymbol{4}\).