1. The sum of the ages of two brothers is 46. The younger brother is 10 more than a third of the older brother's age. How old is the younger brother?
2. You have 59 total coins for a total of $12.05. You only have quarters and dimes. How many of each coin do you have?
y = 10 + x/3
y + x = 46 so y = 46 - x
46 - x = 10 + x/3
36 = 4 x/3
x = 27
y = 46-27 = 19
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25 q + 10 d = 1205
d = 59 - q
so
25 q + 590 - 10 q = 1205
etc
1. To solve this problem, let's first assign variables to represent the ages of the two brothers. Let's call the age of the older brother "x" and the age of the younger brother "y".
According to the problem, the sum of their ages is 46. So we can write the equation: x + y = 46.
The problem tells us that the younger brother is 10 more than a third of the older brother's age. We can represent this with the equation: y = (1/3)x + 10.
Now we have a system of two equations with two variables. We can solve it by substituting the value of y from the second equation into the first equation:
x + ((1/3)x + 10) = 46.
Simplifying the equation, we get: (4/3)x + 10 = 46.
Next, subtract 10 from both sides: (4/3)x = 36.
To solve for x, multiply both sides by 3/4: x = 27.
Now substitute the value of x into either of the original equations to find y:
y = (1/3)(27) + 10 = 9 + 10 = 19.
Therefore, the younger brother is 19 years old.
2. Let's use a systematic approach to solve this problem. Let's assign variables to the number of quarters and dimes. Let's call the number of quarters "q" and the number of dimes "d".
According to the problem, the total number of coins is 59. So we can write the equation: q + d = 59.
The total value of the coins is $12.05. Since there are 100 cents in a dollar, we can convert the total value to cents: $12.05 = 1205 cents.
The value of each quarter is 25 cents and the value of each dime is 10 cents. So we can write the equation: 25q + 10d = 1205.
Now we have a system of two equations with two variables. We can solve it using various methods, such as substitution or elimination.
Let's solve it using substitution. Solve the first equation for q: q = 59 - d. Substitute this expression for q in the second equation:
25(59 - d) + 10d = 1205.
Simplifying the equation, we get: 1475 - 25d + 10d = 1205.
Combine like terms: -15d = -270.
Divide both sides by -15: d = 18.
Now substitute the value of d into the first equation to solve for q:
q = 59 - 18 = 41.
Therefore, you have 41 quarters and 18 dimes.
1. Let's assume the older brother's age as x.
According to the problem, the younger brother's age is 10 more than a third of the older brother's age.
So, the younger brother's age can be represented as (1/3)x + 10.
Now, the sum of their ages is given to be 46.
So, we can write the equation as:
x + (1/3)x + 10 = 46
Combining like terms, we get:
(4/3)x + 10 = 46
Subtracting 10 from both sides, we have:
(4/3)x = 36
Multiplying both sides by 3/4, we get:
x = 27
Therefore, the older brother's age is 27.
To find the younger brother's age, we substitute the value of x in the equation:
Younger brother's age = (1/3)(27) + 10
Younger brother's age = 9 + 10
Younger brother's age = 19
Therefore, the younger brother is 19 years old.
2. Let's assume the number of quarters as x and the number of dimes as y.
The value of a quarter is $0.25, so the total value of the quarters is 0.25x.
The value of a dime is $0.10, so the total value of the dimes is 0.10y.
According to the problem, we have a total of 59 coins in total.
So, we can write the first equation as:
x + y = 59
The total value of all the coins is given as $12.05.
So, we can write the second equation as:
0.25x + 0.10y = 12.05
To solve these equations, we can use substitution or elimination method.
Let's solve by substitution method:
Solve the first equation for x:
x = 59 - y
Substitute the value of x in the second equation:
0.25(59 - y) + 0.10y = 12.05
Distribute and simplify:
14.75 - 0.25y + 0.10y = 12.05
Combine like terms:
-0.15y = -2.70
Divide both sides by -0.15:
y = 18
Substitute the value of y in the first equation:
x + 18 = 59
x = 41
Therefore, there are 41 quarters and 18 dimes.