1) Sarah’s age is ten years more than twice Tanya’s age. The sum of their ages is 64. How old is Sarah?
2) The length of a rectangle is 2 m more than it’s width. The perimeter is 60m. What is its width?
3) A collection of dimes and quarters has a total value of $3.95. If there are 20 coins in the collection, how many are there of each kind?
4) The length of a certain rectangle is 15 cm more than three times its width. If the perimeter of the rectangle is 94 cm, what is the area?
5) The sum of two numbers is 78. If five times the lesser number is subtracted from three times the greater number, the difference is 18. Find the numbers.
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Answer:
Step-by-step explanation:
1. Let Tanya's age be x.
Sarah's age is 10 years more than twice Tanya's age, so Sarah's age is 2x + 10.
The sum of their ages is 64, so we have the equation:
x + (2x + 10) = 64
Simplifying the equation:
3x + 10 = 64
3x = 54
x = 18
Therefore, Tanya's age is 18, and Sarah's age is 2(18) + 10 = 46.
2. Let the width of the rectangle be x.
The length of the rectangle is 2m more than its width, so the length is x + 2.
The perimeter of the rectangle is 60m, so we have the equation:
2(x + x + 2) = 60
Simplifying the equation:
4x + 4 = 60
4x = 56
x = 14
Therefore, the width of the rectangle is 14m, and the length is 16m.
3. Let the number of dimes be x, and the number of quarters be y.
There are 20 coins in total, so we have the equation:
x + y = 20
The total value of the coins is $3.95, so we have the equation:
0.10x + 0.25y = 3.95
We can solve this system of equations by substitution or elimination. For simplicity, we will use substitution.
Solving for x in the first equation:
x = 20 - y
Substituting x in the second equation:
0.10(20 - y) + 0.25y = 3.95
Simplifying the equation:
2 - 0.10y + 0.25y = 3.95
0.15y = 1.95
y = 13
Substituting y in the first equation:
x + 13 = 20
x = 7
Therefore, there are 7 dimes and 13 quarters in the collection.
4. . Let the width of the rectangle be x.
The length of the rectangle is 15cm more than three times its width, so the length is 3x + 15.
The perimeter of the rectangle is 94cm, so we have the equation:
2(x + 3x + 15) = 94
Simplifying the equation:
8x + 30 = 94
8x = 64
x = 8
Therefore, the width of the rectangle is 8cm, and the length is 3(8) + 15 = 39cm.
The area of the rectangle is length times width, so the area is 8cm × 39cm = 312cm².
5. Let the lesser number be x, and the greater number be y.
The sum of the two numbers is 78, so we have the equation:
x + y = 78
If five times the lesser number is subtracted from three times the greater number, the difference is 18, so we have the equation:
3y - 5x = 18
We can solve this system of equations by substitution or elimination. For simplicity, we will use substitution.
Solving for x in the first equation:
x = 78 - y
Substituting x in the second equation:
3y - 5(78 - y) = 18
Simplifying the equation:
8y = 408
y = 51
Substituting y in the first equation:
x + 51 = 78
x = 27
Therefore, the two numbers are 27 and 51.