i have two problems that i need to write an inequality for.
a) One third of a number, decreased by thirty six, is at most twenty two. Find the number.
b) A school's Spanish club is welling bumper stickers. They bought 100 for $55, and have to give 15 cents for every sticker sold. If they plan to sell each sticker for $1.25, how many do they have to sell to make a profit?
Please help and explain so i can understand. Thank you! :)
number = n
one third of number is: n/3
decreased by 36 is: n/3 - 36
is at most 22: n/3 - 36 <= 22
so,
n/3 - 36 <= 22
n/3 <= 58
n <= 174
--------------------
cost for n tickets: 55 + .15n
revenue: 1.25n
profit is when revenue > cost:
1.25n > 55 + .15n
1.1n > 55
n > 50
check:
cost for 50 = 55 + 7.50 = 62.50
revenue for 50 = 1.25 * 50 = 62.50
Sure! Let's break down each problem and write the inequalities.
a) The problem states: "One third of a number, decreased by thirty-six, is at most twenty-two. Find the number."
To write the inequality, we need to start by translating the problem into mathematical terms. The unknown number can be represented as 'x'.
We are given that "One third of the number," so we can write it as (1/3)x.
The problem also states "decreased by thirty-six," so we subtract 36 from (1/3)x.
Finally, the inequality is "is at most twenty-two," so we write it as (1/3)x - 36 ≤ 22.
Now, we can solve the inequality to find the value of x:
(1/3)x - 36 ≤ 22
To isolate x, we first add 36 to both sides of the inequality:
(1/3)x ≤ 22 + 36
Simplifying the right side:
(1/3)x ≤ 58
To get rid of the fraction, we can multiply both sides of the inequality by 3:
3 * (1/3)x ≤ 3 * 58
This simplifies to:
x ≤ 174
Therefore, the solution to the inequality is x ≤ 174. The number must be less than or equal to 174 to satisfy the given conditions.
b) The problem states: "They bought 100 stickers for $55 and have to give 15 cents for every sticker sold. They plan to sell each sticker for $1.25. How many do they have to sell to make a profit?"
To calculate the profit, we need to subtract the total cost from the total revenue.
The total revenue is calculated by multiplying the selling price ($1.25) by the number of stickers to be sold. Let's represent the number of stickers to be sold as 'n'.
Total revenue = $1.25n
The total cost consists of the initial purchase cost ($55) plus the per-sticker cost of 15 cents. The per-sticker cost can be expressed as $0.15n since they have to give 15 cents for every sticker sold.
Total cost = $55 + $0.15n
To make a profit, the total revenue must be greater than the total cost:
$1.25n > $55 + $0.15n
Next, we need to isolate the variable 'n'. First, we subtract $0.15n from both sides of the inequality:
$1.25n - $0.15n > $55
Now, simplify both sides:
$1.10n > $55
Finally, divide both sides by $1.10 to solve for 'n':
n > $55 / $1.10
Simplifying:
n > 50
Therefore, they have to sell more than 50 stickers to make a profit.