Bobby places 12 cards on a table. Each card is either red or black in
colour. If the fraction of black cards is
7/12, What is the smallest number of
additional cards that Bobby needs to put on the table so that the fraction
of black cards becomes 5/8?
Note that the subject is not Ps help. You should put math so a math prof will know to help.
If he adds x black cards and y red cards, we need
(7+x)/(12+x+y) = 5/8
3x = 5y+4
The smallest integer solution to that is x=3, y=1
check:
(7+3)/(12+4) = 10/16 = 5/8
To solve this problem, we need to find the difference between the desired fraction (5/8) and the current fraction (7/12).
Step 1: Find the common denominator
Since the two fractions have different denominators, we need to find a common denominator. In this case, the least common multiple (LCM) of 12 and 8 is 24.
Step 2: Convert the fractions to have the common denominator
Multiply the numerator and denominator of the first fraction (7/12) by 2 to have a common denominator of 24: (7/12) x (2/2) = 14/24
Step 3: Find the difference between the fractions
Now that we have the fractions with a common denominator, we can subtract them to find the difference: 14/24 - 15/24 = -1/24
Step 4: Determine the number of additional cards needed
Since the difference between the fractions is -1/24, we need to add 1/24 of the total number of cards to reach the desired fraction.
Step 5: Calculate the number of additional cards needed
To calculate the number of additional cards needed, we can set up a proportion:
(1/24) / x = 12 / y
Where x represents the number of additional cards needed and y represents the total number of cards after adding the additional cards.
Cross-multiplying gives us:
1y = 12 * 24
y = 12 * 24 / 1
y = 288
So the total number of cards after adding the additional cards is 288.
Step 6: Calculate the number of additional cards needed
To find the number of additional cards needed, substitute the total number of cards (288) into the proportion we set up earlier:
(1/24) / x = 12 / 288
Cross-multiplying gives us:
12x = 1 * 288
x = 288 / 12
x = 24
Therefore, Bobby needs to put an additional 24 cards on the table to have a fraction of 5/8 black cards.
To solve this problem, we need to find out how many additional black cards Bobby needs to put on the table to change the fraction of black cards from 7/12 to 5/8.
Let's analyze the situation step by step. We know that Bobby initially placed 12 cards on the table, and the fraction of black cards is 7/12. This means that out of the 12 cards, 7 are black and 12 - 7 = 5 are red.
To find the number of additional black cards Bobby needs to add, we first need to determine the total number of cards on the table after the addition. Let's represent this total number of cards as x.
After adding the additional black cards, the fraction of black cards becomes 5/8. This means that out of x cards, 5/8 of them are black. Mathematically, we can represent this as:
(5/8) * x = number of black cards
Since we know that the number of black cards was originally 7, we can set up the following equation:
(5/8) * x = 7
To solve for x, we can multiply both sides of the equation by 8/5:
x = 7 * (8/5)
x = 56/5
However, the number of cards should be a whole number, so we need to round up the result to the nearest whole number. Therefore, Bobby needs to have at least 12 + 56/5 ≈ 23.2 cards on the table.
Since Bobby cannot have a fraction of a card, he will need to add at least 24 - 12 = 12 additional cards to ensure the fraction of black cards becomes 5/8.
So, the smallest number of additional cards that Bobby needs to put on the table is 12.