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1. The sum of two numbers is 45. One number is twice the other. What are the two numbers?

2. The sum of three consecutive integers is 72. What are the numbers?

3. Eight years ago, a father was five times as old as his son. Now, he is three times as old. How old are they now?

4. Sasha leaves home biking at 12 mph. Three hours later, her brother Kian starts biking after her at 18 mph. How many hours will it take Kian to catch up to Sasha?

Answer :

  1. To find the two numbers when their sum is 45 and one number is twice the other, let's denote the smaller number as [tex]x[/tex]. Then, the larger number will be [tex]2x[/tex]. According to the problem, their sum is:

[tex]x + 2x = 45[/tex]

Simplifying the equation, we get:

[tex]3x = 45[/tex]

Divide both sides by 3:

[tex]x = 15[/tex]

So, the smaller number is 15, and the larger number is twice that, which is 30. Therefore, the two numbers are 15 and 30.

  1. For the three consecutive integers whose sum is 72, let's denote the first integer as [tex]n[/tex]. The next two consecutive integers will be [tex]n+1[/tex] and [tex]n+2[/tex]. Their sum equation is:

[tex]n + (n + 1) + (n + 2) = 72[/tex]

Simplify the equation:

[tex]3n + 3 = 72[/tex]

Subtract 3 from both sides:

[tex]3n = 69[/tex]

Divide by 3:

[tex]n = 23[/tex]

Therefore, the integers are 23, 24, and 25.

  1. For the ages of a father and son, let's denote the son's current age as [tex]s[/tex] years. Therefore, the father's current age is [tex]3s[/tex]. Eight years ago, their ages were [tex]s - 8[/tex] and [tex]3s - 8[/tex], respectively. At that time, the father was five times as old as the son:

[tex]3s - 8 = 5(s - 8)[/tex]

Simplify and solve the equation:

[tex]3s - 8 = 5s - 40[/tex]

Rearrange to:

[tex]2s = 32[/tex]

Divide by 2:

[tex]s = 16[/tex]

The son is currently 16 years old, and the father is [tex]3 \times 16 = 48[/tex] years old.

  1. For Sasha and Kian's biking situation, Sasha bikes at 12 mph and starts 3 hours before Kian. The distance Sasha covers in those 3 hours is:

[tex]12 \times 3 = 36 \text{ miles}[/tex]

Kian is biking at 18 mph and will be catching up at a relative speed of [tex]18 - 12 = 6 \text{ mph}[/tex]. To catch up to Sasha, he needs to cover the 36 miles difference. The time it takes Kian to cover this distance is:

[tex]\frac{36}{6} = 6 \text{ hours}[/tex]

So, it will take Kian 6 hours to catch up to Sasha.

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