Word Problems Girly Pop
What formula do we use for percent change?
(Final - Initial/ initial) × 100%
Percent change problem: How to solve Cindy starts a job with an annual salary of $120,000. After a year, her annual salary becomes $135,000. What is the percentage increase in her salary?
(Final - Initial/ initial) × 100% (135,000 - 120,000)/120,000 15,000/120,0000 0.125 x 100% 12.5%!
How to solve: Barrel A is made up of a 80% solution of salt water. Barrel B is made up of a 70% solution of salt water. Part of each tank is mixed together with pure water to make a 60% solution of salt water. If ten liters total of solution was made and four liters were taken from Barrel A, how much pure water was added?
-In the final solution, there are 4 L of salt water from barrel A and 4 L of salt water from barrel B. -Thus, the L of pure water added is 10 - 4 - 4 = 2 L.
How to solve: Jamie buys a house which increases in value by 20% over the first year. In the second year, her house decreases by 20% in value from its new worth after the first year. If the value of her house after the second year is $240,000, how much did she originally buy it for?
-Let x be the original cost. -After the first year it increases by 20%: 1.2x -After the second year it decreases by 20% of its current value, making it worth 80% of last year: (0.8)(1.2x) Set equal to current value: (0.8)(1.2x)= 240,000 0.96x= 240,000 x= $250,000
How to solve: There are 66 marbles in a jar with 2 white marbles for each blue marble, and 3 red marbles for each blue marble. How many blue marbles are in the jar?
-Let x equal blue marbles. x + 2x + 3x= 66 6x= 66 x= 11 blue marbles
How do we find the vertex of a parabola?
-Parabolas equations are in the format y= ax² + b + c -To find the vertex we use the equation x= -b/2a to solve for the x value. -Then plug in the x value to solve for y, and that is your point!
How to solve: Brandon finished his last semester of college in which he took 4 courses. He ended the semester with an A in Calculus (4 credits), a C in History (3 credits), a C in Speech (1 credit), and a B in Coding (4 credits). At this college the grades are weighted as follows: an A is 4 grade points, a B is 3 grade points, and a C is 2 grade points. What is Brandon's grade point average for the semester?
-To find GPA we divide the total grade points earned by the total amount of credits. -Multiply the credits by grade points for each class and then add them. -Then divide that by the total credits
How to solve: A classroom of 20 students has an average test grade of 78 points. Two new students were admitted to the class, and after these students took the test, the average of the class was 75 points. What was the score of the last student to take the test if the first newly admitted student scored 45 points?
1. 20x78= 1560 22x75= 1650 2. The sum of the new students scores is 1650-1560= 90, because that's what it changed by. 3. 90-45= 45, this was the last student's score
How to solve: There are 48 total employees that work each day at a 24 hour grocery store. There are always exactly 10 employees working at any given time. What is the average working time each day of all employees?
1. First figure out the total number of hours worked: 24×10= 240 2. Divide total number of working hours by employees: 240/48= 5 hours
How to solve: Vivian drives 84 miles in 2 hours. If her average speed for the first half hour of her trip is 60 mph, then what is her average speed for the remainder of her trip?
1. First through the speed= distance/time equation, find how far she traveled in the first half hour to find how much is left: 60 mph= x miles/0.5 hour x= 30 miles 2. Find how many miles are left: 84-30= 54 miles 3. She traveled 54 miles in 1.5 hours, so let's put it in the equation again: speed= 54 miles/1.5 hours speed= 36 mph
Connie can file 100 papers an hour, while Eric can file 80 papers an hour. Working together, how long will it take them to file 900 papers?
1. If Connie files 100 papers per hour and Eric files 80 papers per hour, together they file: 100+80= 180 papers per hour 2. 180 papers/hour = 900/x x= 900/180 x= 5 hours
How to solve: In a local kickball league, the ratio of college graduates with a graduate degree to non-college graduates is 1:8, and the ratio of college graduates with an undergraduate degree to non-college graduates is 2:3. If one random college graduate is picked, what is the probability that they have a graduate degree?
1. Let's make the ratio numbers that we don't care about the same so we can relate them: 1:8 (we don't care about 8, non college graduates) 2:3 (we don't care about 3 ^) Make it 24: 3:24 16:24 2. Now we can pick off the numbers we do care about and make a little ratio: 3:16, the total being 19 Answer: 3/19
Karen buys enough cans of soda to last her 30 days. The next time Karen is at the store, she sees that the price of cans of soda is on-sale for 25% less than their original value. If Karen spends the exact same amount on cans of soda, how long will the new on-sale supply last her?
1. Lets pretend one soda = $1. We assume she drinks 1 a day so $1x30 days= $30. 2. If the cost of soda is 25% less that would make it only cost $0.75. If she still spends $30 that would mean: 30/0.75= 40 sodas
How to solve: A discount outlet sells product X at a normal price of $12 per unit. The outlet offers the following volume discounts on product X: Order 100-150 units, receive 8 percent discount; 151-200 units, receive 10 percent discount; or order more than 200 units to receive 12 percent discount. Julia saved a discount of $216 on a volume order of product X. How many units of product X did Julia order?
1. Lets start with the 200 unit cutoff for the 10% discount and see how much it is: 200 x $12= $2,400 10% off= $240, too high 2. Lets try the 150 unit cutoff with 8% discount: 150 x $12= $1,800 8% off= $144, too low! 3. Now we know the answer must be between 150 and 200 units.
An organic avocado costs $1.30 more than a regular avocado. If the cost of a regular avocado and an organic avocado together is 300% of the cost of a regular avocado, how much does an organic avocado cost?
1. Make 2 equations with the info they gave us: O= 1.30 + R 3R= O + R 2. Plug in the O equation to solve for R: 3R= (1.30+R) + R 3R= 1.30 +2R R= 1.30 3. O= 1.30 + 1.30 O= $2.60
How to solve: Johnny has 15 toys with a total value of $125. If some of Johnny's toys are worth $5 and the others are worth $10, how many toys does Johnny have that are worth $5?
1. Make 2 equations: x + y= 15 5x + 10y= 125 2. Make y in terms of x and plug in to solve for x: y= 15-x 5x + 10(15-x)= 125 5x + 150 - 10x= 125 -5x= -25 x=5
How to solve: Ariel's dad is 6 years older than her mom. 2 years ago, Ariel's age was a fifth of her mom's current age. If the sum of her parents' ages is 9 times her current age, what is Ariel's current age?
1. Make an equation for each age: D= 6+M A-2= 1/5M D+M= 9A 2. We can solve for A by representing her parents ages in terms of A: M= 5(A-2) Now plug in that for M in the D equation D= 6+ 5(A-2) 3. Now plug in the reworked D and M equations in the last one they are all in: (6+ 5(A-2)) + 5(A-2) = 9A 4. Solve for A= 14!
How to solve: The average weight of three people is 75 kg. If the weight of a fourth person is 95 kg, what is the average weight of the four people together?
1. Make an equation for the 3 people: a+b+c/3 = 75 a+b+c= 3(75) a+b+c= 225 2. Add in the 4th person and solve for the average: a+b+c+95/4 which we know actually means 225+95/4 320/4 = 80kg
How to solve: A jar of spare change has only nickels and pennies in the ratio of 3:5. If the jar contains $2.80 in total, then how many coins are in the jar?
1. Make an equation for the ratio: N/P = 3/5 2. There are 5 cents in one nickel and 1 in every penny, total there are 280 cents. Make an equation of this: 5N + P =280 3. We can write N in terms of P: N= (3/5)P 4. Then we can plug in this N value into the total equation to solve for P (picture). 5. Then we use that value of P, 70, to solve for N. 6. Now we add them together: 70+42= 112
How to solve: Rob drives at 90 km/hour for 2 hours, Sam drives at 88 km/hour for 3 hours and Dave drives at 96 km/hour for 1 hour to finish the trip. What was the average speed of travel for the entire trip?
1. Rob drove 90 for two hours: 90+90= 180 2. Sam drove 88 for three hours: 88+88+88= 264 3. Dave drove 96 for 1 hour. Add them all together and divide by the total number of hours: 180+264+96= 540 km 2+3+1= 6 hours 4. 540/60= 90 km/hour
How to solve: Train A is traveling from Alpha City to Beta City and Train B is traveling from Beta City to Alpha City. Both trains start at the same time and travel toward each other on parallel tracks. The two trains pass each other just as Train A is 7/12 of the way from Alpha City to Beta City. If Train A's speed is 84 miles per hour, which of the following is Train B's speed in miles per hour?
1. Since Train A and B travel for the same amount of time, their distances are proportional to their speeds. 2. If Train A traveled 7/12 of the distance between the two cities, then Train B must have traveled 5/12 of the distance. The ratio of their distances is 7:5, so we can use that proportion to solve for Train B's speed! 7/5 = 84/x 7x= 420 x= 60 mph
How to solve: The water level in a lake drops by 5% after a year. In the second year, it drops an additional 11% from its level after the first year. What is the net percentage drop in water level after 2 years?
1. So we know that after year one the lake is 95% of what it was. 2. In year two it's now 89% of the 95% from year 1! We multiply them together: 0.89 x 0.95= 0.8455 3. Subtract by 1 to get the percentage its dropped by: 1-0.8455= 15.45%
How to solve: A farmer has 144 feet of fence. He wants to enclose a rectangular area so that the width of it is twice as long as the length. What is the maximum area he can enclose?
1. Start out with a formula for the perimeter: 2W + 2L = 144 2. The W= 2L, plug that in: 2(2L) + 2L= 144 6L= 144 L= 24 3. Plug in the L= 24 to solve for the W. W= 2(24) W= 48 4. Area= LW Area= (24)(48) Area= 1152 ft²
How to solve: In a classroom, the ratio of the number of boys to the number of girls is 4:5. If 4 new boys join the class, the ratio of boys to girls becomes 1:1. How many students total are there in the class after the new students join?
1. We can write an expression with x being the number that we multiply by to get the correct numbers: 4x + 4/5x = 1 4x + 4 = 5x 4 = x 2. So the original number of boys was: 4 x 4= 16 The original number of girls was: 5 x 4= 20 16+20+4 boys added= 40 total students
How to solve: Sandra throws a baseball in the air. The path the ball takes in the air follows the equation -5x² + 10x. What is the maximum height the ball reaches during its flight? Assume the ball starts and ends at a height of 0.
1. We need to determine the y-coordinate at the vertex. First we find the x coordinate through the equation x= -b/2a x= -10/(2)(-5) x= 1 2. Now we plug the x value 1 back into the equation to solve for y. x=1 when y was its highest height.
How to solve: A stone is thrown from the top of a cliff. The path of its flight follows the equation -2x² + 8x + 5. If the stone has a horizontal velocity of 4 m/s, how long does it take the stone to reach the highest point of its trajectory?
1. We need to find the vertex, where its at the highest point: x= -b/2a (this gives us the x coordinate). x= -8/(2)(-2) x= 2 2. We don't need the y coordinate and can find the time it takes through this formula: Time= Horizontal distance/Horizontal velocity Time= 2/4 Time= 0.5 sec
An investment in Pear Computers has an initial value of $5,000. A second investment in Macrosoft Computers has an initial value of $7,500. The Pear stock falls by the same percentage as the Macrosoft stock rises. If the new combined investment value is $13,000, by what percentage did the Pear stock fall and the Macrosoft stock rise?
1300= 5,000(1-x) + 7,500(1+x) 1300= 5,000 -5,000x + 7,500 + 7,500x Simplify: x= 0.2 or 20%
1 in = _____ cm
2.54 cm
How to solve: A set of meteor showers called the Areteids occurs every 8 months. If someone sees the Areteids on January 2002, how many more times will the meteors appear before January 2005?
3 years= 36 months 36/8= 4.5 4 times
A hat contains 5 pieces of red paper and 12 pieces of blue paper. How many more pieces of red paper need to be added so that the probability of choosing a piece of red paper is 60%?
5+x/17+x = 0.6 5+x= 0.6(17+x) 5+x= 10.2 + 0.6x 0.4x= 5.2 x=13
How to solve: Dan buys 50 hockey sticks for x dollars each. He sells all of these sticks for 30 dollars each and makes 20% in profit. How much did Dan buy each hockey stick for?
A 20% increase is equal to 1.2 times the original cost: 1.2x = $30 x= $25
How to solve: John receives 1% interest per month on his account, on the 15th day of each month. He makes a payment of $100 on the 20th day each month to pay off his account. If he has a balance of $1,000 on April 10th, how much balance, to the nearest dollar amount, will he have on June 25th?
April 25th: 1,000 + (1000x0.01) -100= $910 May 25th: 910 + (910x0.01) -100= $819.10 June 25th: 819.1 + (819.1x0.001) -100= $727
What is the concentration equation when dealing with combining saline or acids?
Conc= saline or acid/volume
DIRT=
Distance= (rate aka speed) (time)
What is the Compound Interest Formula?
FV = Future Value PV = Present Value r = Interest/growth rate (written as decimal) t = The number of time periods n = Number of times interest the interest is compounded each time period (example: if someone's salary increases once a year, n=1)
Richard is three times as old as Bryan. In six years, Richard will be twice as old as Bryan. How old is Richard right now?
Note: it doesn't say twice as old as Bryan is now, it says will be twice as old as Bryan, as in byan in 6 years as well. 3R = B R+6 = 2(B+6) you know how to solve these
How to solve: Jordan started working for ABC Company in 2013. If his salary increases by 10% every year, and his salary in 2015 is $58,440, what was his starting salary in 2013?
Use the Compound Interest/Growth Formula!
Sam and Joe are the only workers in a food packaging factory. Sam packages 20 kg of food per hour and Joe packages 30 kg of food per hour. Each day they package 300 kg of food. If together they work 12 hours each day, how many hours does Joe work?
We can make two equations!: S= hours Sam works J= hours Joe works S+J= 12 30S + 20J = 300 -You know how to solve these equations lamb!
If a farmer combines 4 L of 25% by volume solution and 4 L of 75% by volume solution, what's the percent by volume of the resulting solution?
We find the volume of solution for each and divide it by the total resulting solution, 8L: 4(0.25)= 1 4(0.75)= 3 1+3= 4 4/8= 50%
What is the Combined work formula and what does it tell us?
t₁= time it took 1st person t₂= time it took 2nd person t total= time would take the two people to do that task together
What is the formula for Dilutions?
● C= Concentration, V = Volume, W = Water ● Dilution = concentration decrease. ● Adding pure water to a concentrated liquid decreases the total concentration (bc pure water's concentration of any other substance is 0%)
