Imagine that there are 3coins in a row. The total value of all the coins together is 35cents. The coins increase in value in the row. Name the coins in the correct order
To solve this problem, let's go step by step.
Step 1: Assign variables for the coins
Let's use "x," "y," and "z" to represent the values of the coins from left to right.
Step 2: Write the given information
We have three coins in a row, and the total value of all the coins is 35 cents. Therefore, we can write the equation:
x + y + z = 35
Step 3: Consider the conditions
We know that the coins increase in value from left to right. So, we need to find a combination of numbers that satisfies this condition.
Step 4: List the combinations
Let's make a table listing all possible values for x, y, and z that satisfy the equation x + y + z = 35.
x | y | z
---------------
1 | 2 | 32
1 | 3 | 31
...
...
...
32 | 33 | 2
33 | 1 | 1
Step 5: Analyze the combinations
Looking at the table, we can see that only one combination satisfies the condition that the coins increase in value. The correct order of the coins is 1 cent, 2 cents, and 32 cents.
So, the coins in the correct order are: 1 cent, 2 cents, and 32 cents.
Let's solve this step-by-step:
Step 1: Let's assume the values of the coins are represented by variables. Let's say the first coin is worth x cents, the second coin is worth y cents, and the third coin is worth z cents.
Step 2: According to the given information, the total value of all the coins together is 35 cents. So we can set up the following equation:
x + y + z = 35
Step 3: The problem states that the coins increase in value in the row. This means the value of z is greater than the value of y, and the value of y is greater than the value of x. Therefore, we can put these inequalities:
x < y < z
Step 4: Now we can solve these equations and inequalities simultaneously to find the values of the coins:
Let's assume the value of the first coin (x) is 10 cents.
Plugging this value into the equation:
10 + y + z = 35
y + z = 25
Step 5: Now, since y is greater than x (10 cents), it must be 15 cents at least. Let's assume y = 15.
Plugging this value into the equation:
15 + z = 25
z = 25 - 15
z = 10
So, the values of the coins in correct order, from left to right, would be 10 cents, 15 cents, and 10 cents.
Considering that the coins increase in value in the row, we can deduce that the first coin must be worth less than the second coin, and the second coin must be worth less than the third coin.
Let's assign variables to the value of the coins. Let's say:
- The value of the first coin is x cents.
- The value of the second coin is y cents.
- The value of the third coin is z cents.
Now, we know that the total value of all three coins together is 35 cents. So we can create the following equation:
x + y + z = 35
Since the coins increase in value, we can also deduce that x < y < z.
To find the correct order of the coins, we need to consider the possible combinations of values for x, y, and z.
Now, let's try different combinations starting with the lowest possible value for x:
1. If x = 1 cent, then y must be at least 2 cents (since the coins increase in value), and z must be at least 3 cents to make a total of 35 cents. However, in this case, the minimum value for the sum x + y + z would be 1 + 2 + 3 = 6 cents, which is greater than 35 cents. So this combination wouldn't work.
2. If x = 2 cents, then y must be at least 3 cents, and z must be at least 4 cents to make a total of 35 cents. Checking the sum x + y + z = 2 + 3 + 4 = 9 cents, we notice that it is nowhere near 35 cents. Hence, this combination is also invalid.
3. If x = 3 cents, then y must be at least 4 cents, and z must be at least 5 cents to reach a total value of 35 cents. Calculating the sum x + y + z = 3 + 4 + 5 = 12 cents, we realize that it is still less than 35 cents. Thus, this combination doesn't work either.
4. If x = 4 cents, then y must be at least 5 cents, and z must be at least 6 cents to sum up to 35 cents. By calculating x + y + z = 4 + 5 + 6 = 15 cents, we observe that this combination is still far from 35 cents.
5. If x = 5 cents, then y must be at least 6 cents, and z must be at least 7 cents to achieve a total value of 35 cents. The sum x + y + z = 5 + 6 + 7 = 18 cents, which is also insufficient.
6. If x = 6 cents, then y must be at least 7 cents, and z must be at least 8 cents to reach a total value of 35 cents. Calculating the sum x + y + z = 6 + 7 + 8 = 21 cents, we find that it is not equal to 35 cents either.
7. If x = 7 cents, then y must be at least 8 cents, and z must be at least 9 cents to make a total of 35 cents. The sum x + y + z = 7 + 8 + 9 = 24 cents, which still doesn't meet the requirement.
8. If x = 8 cents, then y must be at least 9 cents, and z must be at least 10 cents to sum up to 35 cents. By calculating x + y + z = 8 + 9 + 10 = 27 cents, we once again find that it is not equivalent to 35 cents.
As we can see, no matter which value we assign to the first coin, it is impossible for the total value of the three coins to be 35 cents. Therefore, there is no correct order of coins that satisfies the given conditions.