Answer:

3). Let the two numbers be x and y.

Since the sum of the two numbers is 48, we have the equation:

x + y = 48

Now, let's try different combinations of x and y with the given numbers:

a) x = 2, y = 46 (Ratio: 1:23)

b) x = 3, y = 45 (Ratio: 1:15)

c) x = 4, y = 44 (Ratio: 1:11)

d) x = 6, y = 42 (Ratio: 1:7)

e) x = 8, y = 40 (Ratio: 1:5)

f) x = 12, y = 36 (Ratio: 1:3)

g) x = 16, y = 32 (Ratio: 1:2)

h) x = 24, y = 24 (Ratio: 1:1) - Equal numbers not included as they do not form a ratio.

So, the possible ratios that will give the sum of 48 are:

1:23, 1:15, 1:11, 1:7, 1:5, 1:3, and 1:2.

4). Let's solve the problem step by step.

Let the smaller number be 2x, and the greater number be 5x (since the ratio is 5:2).

The ratio of the greater number to the smaller number is given as 5:2:

5x / 2x = 5/2

Their sum is given as 45:

(5x) + (2x) = 45

Combine like terms:

7x = 45

Now, we need to find the correct ratio. To do that, we need to change one of the terms in the ratio. Let's change the ratio to a new ratio of m:n.

Let the new ratio be m:n, where m is the factor we change the greater number (5x), and n is the factor we change the smaller number (2x).

So, the greater number will be m * 5x, and the smaller number will be n * 2x.

Now, we need to find the values of m and n such that the new ratio is correct and their sum is still 45.

The new ratio is:

(m * 5x) : (n * 2x)

We need this ratio to be equal to 5:2, so:

(m * 5x) / (n * 2x) = 5/2

To keep the sum as 45, we need:

(m * 5x) + (n * 2x) = 45

Now, let's solve for m and n:

From the first equation:

(m * 5x) / (n * 2x) = 5/2

Cross-multiply:

2 * (m * 5x) = 5 * (n * 2x)

Simplify:

10mx = 10nx

Now, divide both sides by 10x (since x ≠ 0, we can safely divide):

m = n

From the second equation:

(m * 5x) + (n * 2x) = 45

Since m = n, we can write it as:

(5x + 2x) * m = 45

Combine like terms:

7x * m = 45

Now, divide both sides by 7x (since x ≠ 0, we can safely divide):

m = 45 / 7x

So, the correct ratio would be:

(45 / 7x) : (45 / 7x)

As you can see, the ratio (m:n) is 1:1. This means that both the greater and smaller numbers are equal. Let's find the value of x to get the numbers:

Since the sum is 45:

(45 / 7x) + (45 / 7x) = 45

Combine like terms:

(90 / 7x) = 45

Now, isolate x by multiplying both sides by 7x:

90 = 45 * 7x

Now, divide both sides by 45:

90 / 45 = 7x

Simplify:

2 = 7x

Finally, solve for x:

x = 2/7

Now, we can find the two numbers:

Greater number = 5x = 5 * (2/7) = 10/7

Smaller number = 2x = 2 * (2/7) = 4/7

So, the correct ratio that will make the proportion correct is 10:4 or 5:2. The two numbers are 10/7 and 4/7, which sum up to 45.

5). Let the two numbers be 4x and 3x (since the ratio is 4:3).

The product of the two numbers is given as 192:

(4x) * (3x) = 192

Now, let's solve for x:

12x^2 = 192

Divide both sides by 12:

x^2 = 16

Take the square root of both sides (we consider the positive square root since a length cannot be negative):

x = √16

x = 4

Now that we have the value of x, we can find the two numbers:

First number = 4x = 4 * 4 = 16

Second number = 3x = 3 * 4 = 12

So, the two numbers are 16 and 12.

Step-by-step explanation: