2. Sunita and Ashima's Ages

We are told that Sunita is twice as old as Ashima. Let Ashima's age be $x$, so Sunita's age will be $2x$.

The second condition states that if six years are subtracted from Ashima's age and four years are added to Sunita's age, Sunita will be four times Ashima's age at that time.

This gives us the equation:

$2x + 4 = 4(x - 6)$

Simplifying the equation:

$2x + 4 = 4x - 24$ $4x - 2x = 4 + 24$ $2x = 28$ $x = 14$

So, Ashima is 14 years old, and Sunita is $2 \times 14 = 28$ years old.

To find their ages two years ago:

• Ashima was $14 - 2 = 12$ years old.
• Sunita was $28 - 2 = 26$ years old.

So, two years ago, Ashima was 12 years old, and Sunita was 26 years old.

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3. Twin Prime Numbers

Twin primes are pairs of prime numbers that differ by exactly 2. We are told that the sum of two twin primes is 60.

Let the twin primes be $p$ and $p + 2$.

So, the equation becomes:

$p + (p + 2) = 60$ $2p + 2 = 60$ $2p = 58$ $p = 29$

Thus, the twin primes are $29$ and $31$.

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4. Angles of a Triangle

We are given a triangle where the second angle is one-third of the first angle, and the third angle is 26 degrees more than the first angle. Let the first angle be $x$.

The second angle will be $\frac{x}{3}$, and the third angle will be $x + 26$.

We know that the sum of the angles in a triangle is always 180 degrees, so:

$x + \frac{x}{3} + (x + 26) = 180$

Simplifying the equation:

$x + \frac{x}{3} + x + 26 = 180$ $2x + \frac{x}{3} + 26 = 180$ $2x + \frac{x}{3} = 154$ Multiply through by 3 to eliminate the fraction:

$6x + x = 462$ $7x = 462$ $x = 66$

So, the first angle is 66 degrees. Now, calculate the other two angles:

• The second angle is $\frac{66}{3} = 22$ degrees.
• The third angle is $66 + 26 = 92$ degrees.

Thus, the three angles of the triangle are 66°, 22°, and 92°.

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5. Finding the Number Pairs

We are asked to find pairs of numbers where one number is multiplied by the other, and the result is the sum of the square of the first number and the cube root of the second number.

Let the numbers be $a$ and $b$. The equation is:

$a \times b = a^2 + \sqrt[3]{b}$

This equation involves both multiplication and cube roots, making it complex. To find all pairs that satisfy this condition, it would involve trial and error with different values for $a$ and $b$, or using a computational approach to test multiple possibilities. However, this question is somewhat abstract and lacks more specific constraints (like limits on the numbers).

Given the nature of this question, you could attempt specific values for $a$ and $b$, but a clear pattern is not immediately apparent from basic number theory principles without further exploration.

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Would you like more details on any of the problems or additional explanations? Here are a few related questions for further practice:

1. How would you approach solving systems of linear equations like the one in problem 2 using substitution or elimination?
2. Can you identify other examples of twin primes and their sums?
3. How do you apply the sum of angles property in different types of triangles (isosceles, equilateral, etc.)?
4. Can you solve for the pair of numbers in problem 5 using an algorithm or programming method?
5. How can you check if a pair of numbers satisfies an equation like the one in problem 5 by graphing it?

Tip: Always double-check your units and conditions when solving geometry problems, as small details can change the entire solution!