1. The sum of three numbers is 301. The second number is 3 less than twelve times the first number. The third number is 4 more than seven times the number. Find the three numbers.
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2. Two friends live 15 miles apart. One day they decide to jog and meet each other. Tanya leaves her house and heads east, jogging at a rate of 2.5 miles per hour. At the same time Kelly leaves her house and heads west, jogging at a rate of 3.5 miles per hour. How long does it take for the two friends to meet?
Please help me out with these word problems. Thanks a lot!
1. First number = x, second number = 12x - 3, and if the third number is being compared to the first, it = 7x + 4. The sum of these = 301. Solve for x.
2. 15 = 2.5x + 3.5x
Solve for x.
I hope this helps. Thanks for asking.
1. Let's solve the first question step by step:
Let's assume the first number is x.
According to the question, the second number is 3 less than twelve times the first number, so the second number would be 12x - 3.
Similarly, the third number is 4 more than seven times the first number, so the third number would be 7x + 4.
Now we know the sum of the three numbers is 301. We can write an equation to represent this:
x + (12x - 3) + (7x + 4) = 301
Now, let's simplify and solve for x:
20x + 1 = 301
20x = 300
x = 15
So the first number (x) is 15.
To find the second number, substitute x = 15 into the equation:
12x - 3 = 12(15) - 3 = 177
To find the third number, substitute x = 15 into the equation:
7x + 4 = 7(15) + 4 = 109
So the three numbers are 15, 177, and 109.
2. Now let's solve the second question:
The distance Tanya covers is 2.5 miles per hour and Kelly covers is 3.5 miles per hour. They are covering the total distance of 15 miles.
To find the time it takes for them to meet, we can calculate the total time for each person and take the maximum. Since they are traveling towards each other, it will be the same time for both of them.
Let's assume the time it takes for them to meet is t.
For Tanya, the distance she covers is 2.5 miles/hour, so the time she takes is given by:
Time taken by Tanya = Distance / Speed
t = 15 / 2.5
For Kelly, the distance she covers is 3.5 miles/hour, so the time she takes is given by:
Time taken by Kelly = Distance / Speed
t = 15 / 3.5
Now, we need to find the maximum of these two times. Let's calculate both:
t_Tanya = 15 / 2.5 = 6 hours
t_Kelly = 15 / 3.5 = 4.2857 hours (rounded to 4 decimal places)
Therefore, the two friends will meet after approximately 6 hours.
I hope this explanation helps you understand the process of solving word problems. If you have any further questions, feel free to ask!