I thought of a number and multiplied it by 2 1/2, divided the result by 1 1/5, subtracted 7/18 from it, and got 1 5/6. What was my number?
Let's solve this step by step:
First, let's represent the unknown number as "x."
The problem states that the number was multiplied by 2 1/2, which can be written as 5/2. Multiplying the unknown number by 2 1/2 gives us (5/2) * x, which simplifies to (5/2)x.
Next, the result is divided by 1 1/5, which can be written as 6/5. Dividing (5/2)x by 6/5 gives us [(5/2)x] / (6/5). We can simplify this expression by multiplying by the reciprocal, so [(5/2)x] * (5/6), which becomes (25/12)x.
Now, 7/18 is subtracted from (25/12)x. Therefore, our equation becomes (25/12)x - 7/18 = 1 5/6.
Next, we need to convert 1 5/6 into a fraction. 1 can be rewritten as 6/6, so 1 5/6 becomes (6/6) + 5/6 = 11/6.
Now, we have the equation (25/12)x - 7/18 = 11/6.
To clear the fractions, we can multiply every term by the least common denominator (LCD) of 12, 18, and 6, which is 36. Multiplying both sides of the equation by 36 gives us:
36 * [(25/12)x - 7/18] = 36 * (11/6)
[(36 * 25/12)x] - [(36 * 7/18)] = 36 * 11/6
[(3 * 25/12)x] - [(2 * 7/18)] = 2 * 11/1
(75/12)x - (14/18) = 22
To simplify further, we can convert all fractions to have the same denominator of 36:
(75/12)x - (14/18) = 22
(75/12)x - (28/36) = 22
(75/12)x - (14/9) = 22
Next, to eliminate any decimals, we can multiply every term by 12:
12 * [(75/12)x - (14/9)] = 12 * 22
[(12 * 75/12)x] - [(12 * 14/9)] = 12 * 22
75x - (14 * 12/9) = 264
75x - (168/9) = 264
Now, we convert 264 to have a denominator of 9:
75x - (168/9) = (264 * 9/9)
75x - (168/9) = 2376/9
To simplify further, we multiply every term by 9:
9 * [75x - (168/9)] = 9 * (2376/9)
(9 * 75x) - (9 * 168/9) = 2376/1
675x - 168 = 264
Next, we isolate the term with x by moving the constant term to the other side of the equation:
675x - 168 = 264
675x = 264 + 168
675x = 432
Now, to find the value of x, we divide both sides of the equation by 675:
x = 432/675
After simplifying the fraction, we find that x = 16/25.
Hence, the number you thought of was 16/25.
Wrong.
Apologies for the mistake. Let's solve it again:
Let's represent the unknown number as "x."
The problem states that the number was multiplied by 2 1/2, which can be written as 5/2. Multiplying the unknown number by 2 1/2 gives us (5/2) * x, which simplifies to (5/2)x.
Next, the result is divided by 1 1/5, which can be written as 6/5. Dividing (5/2)x by 6/5 gives us [(5/2)x] / (6/5). We can simplify this expression by multiplying by the reciprocal, so [(5/2)x] * (5/6), which becomes (25/12)x.
Now, 7/18 is subtracted from (25/12)x. Therefore, our equation becomes (25/12)x - 7/18 = 1 5/6.
Next, we need to convert 1 5/6 into a fraction. 1 can be rewritten as 6/6, so 1 5/6 becomes (6/6) + 5/6 = 11/6.
Now, we have the equation (25/12)x - 7/18 = 11/6.
To clear the fractions, we can multiply every term by the least common denominator (LCD) of 12, 18, and 6, which is 36. Multiplying both sides of the equation by 36 gives us:
36 * [(25/12)x - 7/18] = 36 * (11/6)
[(36 * 25/12)x] - [(36 * 7/18)] = 36 * 11/6
[(3 * 25/12)x] - [(2 * 7/18)] = 2 * 11/1
(75/12)x - (14/18) = 22
To simplify further, we can convert all fractions to have the same denominator of 36:
(75/12)x - (14/18) = 22
(75/12)x - (28/36) = 22
(75/12)x - (14/9) = 22
Next, to eliminate any decimals, we can multiply every term by 12:
12 * [(75/12)x - (14/9)] = 12 * 22
[(12 * 75/12)x] - [(12 * 14/9)] = 12 * 22
75x - (14 * 12/9) = 264
75x - (168/9) = 264
Now, we convert 264 to have a denominator of 9:
75x - (168/9) = (264 * 9/9)
75x - (168/9) = 2376/9
To simplify further, we multiply every term by 9:
9 * [75x - (168/9)] = 9 * (2376/9)
(9 * 75x) - (9 * 168/9) = 2376/1
675x - 168 = 264
Next, we isolate the term with x by moving the constant term to the other side of the equation:
675x - 168 = 264
675x = 264 + 168
675x = 432
Now, to find the value of x, we divide both sides of the equation by 675:
x = 432/675
However, upon reviewing the question, I mistakenly calculated the steps involved in the equation. I apologize for the error.
Let's solve it correctly:
First, we solve the equation step by step:
Let's say the unknown number is "n."
The problem states that the number was multiplied by 2 1/2, which can be written as 5/2. Multiplying the unknown number by 2 1/2 gives us (5/2) * n, which simplifies to (5/2)n.
Next, the result is divided by 1 1/5, which can be written as 6/5. Dividing (5/2)n by 6/5 gives us [(5/2)n] / (6/5). We can simplify this expression by multiplying by the reciprocal, so [(5/2)n] * (5/6), which becomes (25/12)n.
Now, 7/18 is subtracted from (25/12)n. Therefore, our equation becomes (25/12)n - 7/18 = 1 5/6.
Next, we need to convert 1 5/6 into a fraction. 1 can be rewritten as 6/6, so 1 5/6 becomes (6/6) + 5/6 = 11/6.
Now, we have the equation (25/12)n - 7/18 = 11/6.
To clear the fractions, we can multiply every term by the least common denominator (LCD) of 12, 18, and 6, which is 36. Multiplying both sides of the equation by 36 gives us:
36 * [(25/12)n - 7/18] = 36 * (11/6)
[(36 * 25/12)n] - [(36 * 7/18)] = 36 * 11/6
[(3 * 25/12)n] - [(2 * 7/18)] = 2 * 11/1
(75/12)n - (14/18) = 22
To simplify further, we can convert all fractions to have the same denominator of 36:
(75/12)n - (14/18) = 22
(75/12)n - (28/36) = 22
(75/12)n - (14/9) = 22
Next, to eliminate any decimals, we can multiply every term by 12:
12 * [(75/12)n - (14/9)] = 12 * 22
[(12 * 75/12)n] - [(12 * 14/9)] = 12 * 22
75n - (14 * 12/9) = 264
75n - (168/9) = 264
Now, we convert 264 to have a denominator of 9:
75n - (168/9) = (264 * 9/9)
75n - (168/9) = 2376/9
To simplify further, we multiply every term by 9:
9 * [75n - (168/9)] = 9 * (2376/9)
(9 * 75n) - (9 * 168/9) = 2376/1
675n - 168 = 264
Next, we isolate the term with n by moving the constant term to the other side of the equation:
675n - 168 = 264
675n = 264 + 168
675n = 432
Now, to find the value of n, we divide both sides of the equation by 675:
n = 432/675
Upon re-checking the solution, it seems that I have made another error in the calculation. I apologize for the confusion and I am providing an updated solution.
To solve the equation (25/12)n - 7/18 = 1 5/6, we need to convert 1 5/6 to an improper fraction.
1 5/6 can be rewritten as 11/6.
Now our equation becomes (25/12)n - 7/18 = 11/6.
To find a common denominator, we take the least common multiple of 12 and 18, which is 36.
Multiplying all terms by 36:
(25/12)n * 36 - 7/18 * 36 = 11/6 * 36
Canceling out the denominators:
25n * 3 - 7 * 2 = 11 * 6
75n - 14 = 66
Adding 14 to both sides of the equation:
75n = 66 + 14
75n = 80
Finally, we divide both sides of the equation by 75:
n = 80/75
Simplifying the fraction gives:
n = 16/15
Therefore, the number you thought of is 16/15.
Anna bought 2 3/5 lb of grapes, and Sanika bought 1/2 lb less than Anna. How many pounds of grapes did they buy in all?
To find the total weight of grapes bought by Anna and Sanika, we need to add their weights together.
Anna bought 2 3/5 lb of grapes, which can be written as 13/5 lb.
Sanika bought 1/2 lb less than Anna's weight, so we subtract 1/2 lb from Anna's weight:
13/5 lb - 1/2 lb
To subtract fractions, we need a common denominator. The least common multiple of 5 and 2 is 10, so we convert both fractions to have a denominator of 10:
(13/5) * (2/2) - 1/2
This gives us:
26/10 - 1/2
To subtract fractions, we need a common denominator of 10:
(26/10 - 5/10) lb
This simplifies to:
21/10 lb
Now, to find the total weight of grapes bought, we add Anna's weight to Sanika's weight:
(13/5 lb) + (21/10 lb)
To add fractions, we need a common denominator of 10:
(26/10 lb) + (21/10 lb)
This gives us:
47/10 lb
Therefore, Anna and Sanika bought a total of 47/10 lb of grapes, which can be simplified to 4 7/10 lb.
To find the number you started with, we can work backwards through the given steps. Let's break it down step-by-step:
1. Before subtracting 7/18 from it, the result was 1 5/6.
2. Thus, to find the number before the subtraction, we need to add 7/18 to 1 5/6.
To add fractions, we need to find a common denominator. The common denominator for 6 and 18 is 18.
Converting 1 5/6 to an improper fraction:
1 5/6 = (1 * 6 + 5)/6 = 11/6
Adding 7/18 to 11/6:
(11/6) + (7/18) = (11 * 3/6 * 3) + 7/18
= (33/18) + (7/18) = (33 + 7)/18 = 40/18
3. Before dividing the result by 1 1/5, the number was 40/18.
Next, to reverse the division by 1 1/5, we need to multiply the result by 1 1/5 reciprocal (flipping the fraction).
The reciprocal of 1 1/5 is 5/6.
Multiplying 40/18 by 5/6:
(40/18) * (5/6) = (40 * 5)/(18 * 6) = 200/108
4. Before multiplying the number by 2 1/2, the result was 200/108.
Lastly, to undo the multiplication by 2 1/2, we need to divide the result by 2 1/2 reciprocal.
The reciprocal of 2 1/2 is 2/5.
Dividing 200/108 by 2/5:
(200/108) รท (2/5) = (200/108) * (5/2)
= (200 * 5)/(108 * 2) = 1000/216
5. Therefore, the number you started with was 1000/216, which can be simplified to 125/27.
So, your number was 125/27.
To find the original number, we need to reverse the operations that were performed.
Let's break down the given sequence of operations step by step:
1. "Multiplied it by 2 1/2": This means you took your number and multiplied it by 2.5 (or 5/2).
2. "Divided the result by 1 1/5": This means you took the previous result and divided it by 1.2 (or 6/5).
3. "Subtracted 7/18 from it": This means you took the previous result and subtracted 7/18.
After performing these operations, you ended up with the number 1 5/6.
To find the original number, we need to reverse the operations one by one:
1. Add 7/18 to 1 5/6: Since we want to subtract 7/18, we need to add its negation, which is -7/18. Adding 1 5/6 and -7/18 gives us 1 5/6 + (-7/18) = 1 5/6 - 7/18.
2. Multiply the result by 1 1/5: We need to find the reciprocal of 1 1/5, which is 5/6, and multiply it by the previous result to get the original number. So, (1 5/6 - 7/18) * (5/6) = (11/6 - 7/18) * (5/6).
3. Divide the result by 2 1/2: We need to find the reciprocal of 2 1/2, which is 2/5, and divide it by the previous result to get the original number. So, [(11/6 - 7/18) * (5/6)] / (2/5).
Now, let's perform these calculations step by step:
Calculating the result of 1 5/6 - 7/18:
1 5/6 - 7/18 = (18/18 * 1) + (15/18) - (7/18) = 33/18 - 7/18 = 26/18 = 13/9.
Calculating the result of (13/9) * (5/6):
(13/9) * (5/6) = (13 * 5) / (9 * 6) = 65/54.
Calculating the result of (65/54) / (2/5):
[(65/54) / (2/5)] = (65/54) * (5/2) = (65 * 5) / (54 * 2) = 325/108.
Therefore, your original number is 325/108.