Steven bought a piece of meat and cut it into three pieces. The mass of the first piece of meat was 5/8 of the mass of the second piece of meat. The mass of the second piece of meat was 4/7 of the mass of the third piece of meat.
a. Find the ratio of the mass of the first piece of meat to that of the second piece of meat to that of the third piece of meat.
b. What is the ratio of the mass of the second piece of meat to the total mass of the three pieces of meat?
c. What fraction of the total mass of the three pieces of meat was the mass of the third piece of meat?
To solve this problem step-by-step:
Step 1: Assign variables
Let's assign variables to represent the masses of the three pieces of meat. Let's say:
- The mass of the first piece of meat is x.
- The mass of the second piece of meat is y.
- The mass of the third piece of meat is z.
Step 2: Express the given information as equations
From the problem statement, we have the following information:
- The mass of the first piece of meat is 5/8 of the mass of the second piece of meat: x = (5/8)y.
- The mass of the second piece of meat is 4/7 of the mass of the third piece of meat: y = (4/7)z.
Step 3: Solve for the variables
We can use the given equations to solve for x, y, and z.
From x = (5/8)y, we can multiply both sides by 8/5 to get:
8/5 * x = y.
From y = (4/7)z, we can multiply both sides by 7/4 to get:
7/4 * y = z.
So the ratio of the masses of the three pieces of meat is:
x : y : z = x : (8/5 * x) : (7/4 * 8/5 * x) = 1 : 8/5 : 7/4 * 8/5.
Step 4: Simplify the ratio
To simplify the ratio, we can convert the fractions into decimals and find the least common multiple of the denominators to get rid of fractions:
1 : (8/5) : (56/20) = 1 : 1.6 : 2.8.
Therefore, the ratio of the masses of the three pieces of meat is 1 : 1.6 : 2.8.
Step 5: Find the ratio of the second piece of meat to the total mass
To find the ratio of the mass of the second piece of meat to the total mass of the three pieces of meat, we need to add up the masses of all three pieces of meat:
Total mass = x + y + z.
Using the equations x = (5/8)y and y = (4/7)z, we can substitute these values into the expression for the total mass:
Total mass = (5/8)y + y + (7/4) * (5/8)y.
Simplify the expression:
Total mass = (5/8 + 1 + 35/32)y = (77/32)y.
So the ratio of the mass of the second piece of meat to the total mass is:
y : (77/32)y = 1 : (77/32) = 32:77.
Therefore, the ratio of the mass of the second piece of meat to the total mass of the three pieces of meat is 32 : 77.
Step 6: Find the fraction of the total mass represented by the third piece of meat
To find the fraction of the total mass represented by the third piece of meat, we need to compare the mass of the third piece (z) to the total mass (Total mass = (77/32)y).
The fraction representing the mass of the third piece is:
(z / Total mass) = (z / (77/32)y).
Since y = (4/7)z, we can substitute this value into the expression:
(z / (77/32)y) = (z / (77/32) * (4/7)z).
Cancel out the z terms:
(z / (77/32) * (4/7)z) = (1 / (77/32) * (4/7)).
Simplify the expression:
(1 / (77/32) * (4/7)) = (1 * 32 / 77 * 4/7) = 8/77.
Therefore, the fraction of the total mass represented by the third piece of meat is 8/77.
To solve this problem, we'll use algebra to represent the relationships between the masses of the three pieces of meat. Let's start by assigning variables to the unknowns.
Let:
- Mass of the first piece of meat be x
- Mass of the second piece of meat be y
- Mass of the third piece of meat be z
a. To find the ratio of the masses, we need to express each piece of meat's mass in terms of one another. According to the given information:
- The mass of the first piece (x) is 5/8 of the mass of the second piece (y).
So, we can write: x = (5/8) * y.
- The mass of the second piece (y) is 4/7 of the mass of the third piece (z).
So, we can write: y = (4/7) * z.
We can now solve these two equations to find the values of x, y, and z.
Solving the first equation for y, we get: y = (8/5) * x.
Substituting this into the second equation, we get: (8/5) * x = (4/7) * z.
To solve for the variables, we can equate the ratios of each variable: (8/5) * x = (4/7) * z.
Cross multiplying, we have: 8z = 20x.
Now, we can express the ratio of the three masses using the equation derived: x : y : z = x : (8/5) * x : 20x.
Simplifying this, we get: x : y : z = 1 : 8/5 : 20/1.
The ratio of the masses of the three pieces of meat is 1 : 8/5 : 20/1, or simplified, 1 : 8/5 : 20.
b. To find the ratio of the mass of the second piece of meat to the total mass of the three pieces, we need to determine the total mass.
The total mass is given by the sum of the masses of the three pieces: x + y + z.
Substituting the values we found earlier into this equation, we have:
x + (8/5) * x + 20x = (5/5) * x + (8/5) * x + (100/5) * x,
which simplifies to: 29x = 14x.
Dividing both sides by 14x, we find: 29x/14x = 14x/14x,
which simplifies to: 29/14 = 1.
So, the total mass of the three pieces of meat is 1.
The ratio of the mass of the second piece of meat to the total mass of the three pieces is therefore 8/5 : 1, or simplified, 8 : 5.
c. Finally, to find the fraction of the total mass of the three pieces that the mass of the third piece represents, we can express it as a ratio.
The third piece of meat has a mass of z, and the total mass is 1. So, the fraction can be written as: z/1.
Since z represents the entire mass of the third piece, the fraction is simply z.
Therefore, the fraction of the total mass of the three pieces of meat that the mass of the third piece represents is z.
third piece ---- x
2nd piece = (4/7)x
1st piece = (5/8)(4/7)x = (5/14)x
a) ratio of 1st : 2nd : third
= (5/14)x : (4/7) : x
= 5/14 : 4/7 : 1
= 5 : 8 : 14
b) do the same way
c) add up the 3 pieces
x + (4/7)x + (5/14)x
= (14x + 8x + 5x)/14 = (27/14)x
first : total = (5/14) x : (27/14) x
= 5 : 27
check my arithmetic