Mr. Badger bought 12 pens every month until their price rose to 15 cents each. Now he can only afford to buy 10 pens at the same total cost as before. What was the original price of each pen?
There are 41 pigs and chickens on a farm. If there are 100 legs counted altogehter, how many of each animal are there?
One day, 5/6 of the office staff came to work. If three more had been away, this fraction would have decreased to 3/4. How many people are on full staff?
Mrs. Grant was 20 when her eldest child, tess, was born. Carly was born 2 years later and Troy another 4 years later. Now the average of their four ages is 39. How old are Mrs. Grant and her three children?
Please help...
2. We have X pigs and Y chickens.
Eq1: X + Y = 41 Pigs and chicken.
Each pig has 4 legs and each chicken has 2 legs:
Eq2: 4X + 2Y = 100 Legs.
Multiply both sides of Eq1 by -2 and
add the 2 Eqs:
-2X - 2Y = - 82,
4X + 2Y = 100,
2X = 18,
X = 9 Pigs.
Substitute 9 for X in Eq1:
9 + y = 41,
Y = 41 - 9 = 32 Chickens.
3. 5X/6 - 3 = 3X/4,
Multiply both sids by 12:
10x - 36 = 9X,
10X - 9X = 36,
X = 36 Staff members.
4. Tess: X Years old.
Mrs. Grant: X+20.
Carly: X-2.
Troy: X-6.
(X + (X+20) + (X-2) + (X-6)) / 4 = 39,
(4X + 12) / 4 = 39,
Multiply both sides by 4:
4X + 12 = 156,
4X = 156 - 12 = 144,
X = 36 = Tess's age.
X+20 = 36+20 = 56 = Mrs. Grant's age.
X-2 = 36-2 = 34 = Carly's age.
X-6 = 36-6 = 30 = Troy's age.
To solve these types of problems, we can use algebraic equations or logical reasoning. I will explain how to solve each question step by step.
1. Mr. Badger bought 12 pens every month until their price rose to 15 cents each. Now he can only afford to buy 10 pens at the same total cost as before. What was the original price of each pen?
Let's assume the original price of each pen was x cents. If Mr. Badger bought 12 pens every month, the total cost would be 12x cents. Since he can now only afford to buy 10 pens at the same total cost, the new price must be 15 cents per pen.
We can set up an equation to solve for x:
12x = 10 * 15
12x = 150
x = 150 / 12
x = 12.5
So, the original price of each pen was 12.5 cents.
2. There are 41 pigs and chickens on a farm. If there are 100 legs counted altogether, how many of each animal are there?
Assuming that each pig has 4 legs and each chicken has 2 legs, we can set up two equations to represent the total number of animals and the total number of legs.
Let's assume the number of pigs is p and the number of chickens is c.
We have two conditions: p + c = 41 (equation for the total number of animals) and 4p + 2c = 100 (equation for the total number of legs).
We can solve these two equations simultaneously to find the values of p and c.
From the first equation, we can express p in terms of c as p = 41 - c.
Substituting this into the second equation, we get: 4(41 - c) + 2c = 100.
Expanding and simplifying, we have: 164 - 4c + 2c = 100.
Combine like terms: -2c = -64.
Dividing both sides by -2, we find: c = 32.
Substituting this value back into the first equation, we have: p + 32 = 41.
Then, p = 41 - 32 = 9.
Therefore, there are 9 pigs and 32 chickens on the farm.
3. One day, 5/6 of the office staff came to work. If three more had been away, this fraction would have decreased to 3/4. How many people are on full staff?
Let's assume the total number of people on full staff is x.
According to the problem, on the given day, 5/6 of the staff came to work, which means (5/6)x people came to work. If three more people had been away, the fraction would have decreased to 3/4. Therefore, (5/6)x - 3 people came to work.
Now we can set up an equation to solve for x:
(5/6)x - 3 = (3/4)x
We can now solve for x by manipulating the equation:
Multiply both sides of the equation by 12 to eliminate the fractions:
10x - 36 = 9x
Subtract 9x from both sides:
x - 36 = 0
x = 36
Therefore, there are 36 people in the full staff.
4. Mrs. Grant was 20 when her eldest child, Tess, was born. Carly was born 2 years later, and Troy was born another 4 years later. Now the average of their four ages is 39. How old are Mrs. Grant and her three children?
Let's denote Mrs. Grant's age as G and the ages of her three children as T, C, and R.
We can set up the following equations based on the given information:
T = G - 20 (Tess's age)
C = T + 2 (Carly's age, 2 years older than Tess)
R = C + 4 (Troy's age, 4 years older than Carly)
The average of their four ages is 39, so we can express this as an equation:
(G + T + C + R) / 4 = 39
To simplify, substitute the expressions for T, C, and R into the average equation:
(G + (G - 20) + ((G - 20) + 2) + (((G - 20) + 2) + 4)) / 4 = 39
Expanding and simplifying, we have:
(4G - 20 + 4G - 20 + 6G - 44) / 4 = 39
(14G - 84) / 4 = 39
14G - 84 = 156
14G = 240
G = 240 / 14
G ≈ 17.14
Since ages are typically whole numbers, we round G to the nearest whole number. Mrs. Grant's age is approximately 17 years old.
Now, substitute G = 17 into the equations to find the ages of the children:
T ≈ 17 - 20 ≈ -3
C ≈ -3 + 2 ≈ -1
R ≈ -1 + 4 ≈ 3
Since negative ages do not make sense, we disregard T and C. We conclude that the oldest child, Tess, is 20 years old and the youngest child, Troy, is 3 years old.