Math 2: Problem Solving. Steps . Read the problem (try to understand it upon the first read) Determine the type of problem Identify the variables Set up the equation Solve the equation/s Find the final solution Check the solution *Use trial and error only if you think it will be faster*
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*Use trial and error only if you think it will be faster*
(The examples in the following slides emphasize the equations and not the solutions)
A1Q1 + A2Q2 + A3Q3 + .... = Total Amount
Where A= value of coins and Q = quantity of coins
Example: Percy has a box in his tent that contains 45 coins. Some are nickels and the rest are dimes. If the total value of the coins is $3.25, how many of each coin does he have?
Let x= # of nickels and 45-x = # of dimes
5(x) + 10(45-x) =325
X = 25and 45-x = 20
Final Solution: Percy has 25 nickels and 20 dimes in his box
Percentage = Rate x Base
Where Percentage = Part of the whole
Rate = usually with the % or in decimal form
Base = whole or principal amount OR
Interest = Percentage x Rate x Time
Where Principal = amount originally invested or loaned
Rate = interest rate of the bank or institution
Time= how long the money is invested
Jesun Burn earned $8,000 last year by selling fish balls. He invested part at 2% simple interest and the rest at 3%. He earned a total of $210 in interest. How much did he invest at each rate?
(Interest at 2%) + (Interest at 3%) = total interest
0.02x + 0.03(8,000-x) = $210
X = 3,000
8,000-x = 5,000
Final answer: $3,000 and $5,000 were invested at 2% and 3% respectively
Amount after one year = Principal x (1+Rate)
What is the interest compounded per annum on a principal of P6,000 when the amount after one year is P7,200?
Let x = rate or the interest compounded per annum
7,200 = 6,000(1+x)
x= 0.2 or 20%
Final answer: The interest compouned per annum is 20%
In decimal, % = new price – original price
If the % is positive, the price increased
If the % is negative, the price increased
A belt costs P650 but Chloe was able to purchase it at a price of P601.25. What was the percentage of the decrease?
% = 601.25-650 / 650 = -0.075 or 7.5
Final answer: The percentage decrease was 7.5%
Discount = %Discount x (Original Price)
Discounted Price = Original Price – Discount
New Price = (1-%Discount) x Original Price
Gustongbumilini Stark ngpitakananagkakahalagang P475. Kung may tawadna 20%, magkanoangkanyangbabayarin?
Discount = 0.20(475) = P95
Discounted price = P475-P95 = P380
Final answer: Angbabyarinni Stark at P380
Commission = %Commission x Total Earnings
A car agent sells a brand new car for P283,000. If he is given 10% commission, how much will he receive?
Commission = 0.1(283,000)
Final answer: The car agent will receive P28,300 in commission
Tax = %Tax x (Original Price)
Price with tax = Original Price + Tax
New Price = (1+%Tax) x Original Price
20% tax is imposed on purchasing a pair of rare sandals. If the original price of the sandals is P35,000 how much will one have to pay upon buying?
Tax = 0.2(35,000) = P7,000
P35,000 + 7,000 = 42, 000
Final answer: One has to pay P42,000 for the shoes
A1Q1 + A2Q2 + A3Q3 + .... = Total Amount
%origQorig + %mixQmix = %final x (Qorig + Qmix)
Removal of solution
%origQorig - %mixQmix = %final x (Qorig - Qmix)
Replacing of a solution
%origQorig - %origQorig + %mixQmix = %finalQorig
How many liters of a 25% alcohol solution should Zyrene mix with 20 liters of a 55% solution to get a 30% solution?
0.55(20) + 0.25x = 0.30(20+x)
X = 100 liters
Final answer: She should have 100 liters of the 25% solution
There is no general formula. Derive the equations based on how they are stated in the problem. Use direct translation of words into math symbols.
If x is the present age then
The age n years ago is x-n
The age m years from now is x+m
Helen is 5 years younger than Paris. 10 years from now, Paris’ age will be 50 years less than twice Helen’s age. How old is Paris now?
Let P = Paris’ present age; H = Helen’s present age
Present: H = P-5; 10 years from now: P + 10 = 2(H+10) – 50
P + 10 = 2(P-5+10)-50; P = 50
Final answer: Paris is 50 years old now
A fraction is a ratio of 2 numbers. If the value of the fraction is 2/3, the numerator : denominator is not only 2:30 but also 1:1.5 or 30:45... Etc. Thus, the numerator may be represented by 2x, the denominator by 3x, and the fraction by 2x/3x.
The sum of the numerator and denominator of a fraction is 16. If the value of the fraction is 1/3, what is the denominator?
Let x / 3x = original fraction, where x is the common factor.
X + 3x = 16; x = 4
Hence, 3x = 12
Final answer: The denominator is 12
Consecutive integers are one unit apart and can be represented by x, x+1, and x+2 and so on
Consecutive even or odd numbers are 2 units apart and can be represented by x, x+2, x+4, and so on
Ang sum ngtatlongmagkakasunodna even integer ay 84. Anoangikalawang integer?
x + (x+2) + (x+4) = 84
x = 26; x + 2 = 28
Final answer: Ikalawang even integer = 28
A 3-digit number whose hundreds digit is x, tens digit is y, and units digit is z can be expressed as 100x + 10y + z. The reverse of this number is written as 100z + 10y + x.
The sum of the digits of a two-digit number is 12. If the digits are reversed, the new number is 54 more than the original number. Find the two-digit number.
Let x = tens digit; y = units digit 16. If the value of the fraction is 1/3, what is the denominator?
10x + y = original number
10y + x = reversed number
Equation 1: x + y = 12
Equation 2: new = original + 54
10y + x = 10x + y + 54
y - x = 6
y + x = 12
y – x = 6
2x = 6
x = 3 and 3 + y = 12 so y 9
Final answer: the original number is 39
Distance = Rate x Time
Motion in Opposite directions
D1 + D2 = Dtotalor R1T1 + R2T2 = Dtotal
Motion in Same direction (catch-up)
D1 = D2 or R1T1 = R2T2
Dgoing = Dreturning or RgoTgo = RretTret
Mica drives her car and it takes her 30 minutes to reach her work. As she returns, she rides the bus and it takes her 45 minutes to reach her home. The average speed of the bus is 12 miles per hour less than her speed when driving. Find the distance she travels to work.
Let x = Mica’s speed when driving
½ x = ¾(x-12)
X = 36 mph
D = ½ x = 18 miles
Final answer: Mica drives a distance of 18 miles to her work
x = rate of the vehicle in still current
y = rate of the current
With the wind or downstream
x + y
Against the wind or upstream
x - y
It takes 3 hours for a boat to travel 75 miles downstream. The same boat can travel 60 miles upstream in 4 hours. Find the speed of the boat in still water and the rate of the current.
X = rate of the boat in still water
Y = rate of the current
x + y = 25
x – y = 15
2x = 40
x = 20 mph
Rate = Distance / Time
x + y = 75/3
x – y = 60/4
The rate of the current is 5 mph, the speed of the boat is 20 mph
Work done = Rate x Time
Work done together
(R1 + R2) T = W
Work done in opposition
(Rfaster– Rslower) T = W
An inlet pipe can fill a tank of water in 6 hours while an outlet pipe can drain the same tank in 8 hours. If a half-full tank is first drained in two hours and is filled while the outlet pipe is still open, how long should the inlet pipe be opened to fill the tank?
x / 6 – (x+2) / 8 = 1 / 2 (tank)
X = 18
The inlet pipe should be opened for 18 hours
Using the 12 o clock position as a reference point, we subdivide the clock into 60 equal “arcs.”
When the minute-hand moves 60 units, the hour-hand moves 5 units. As the minute hand moves M units, the hour hand moves 5M/60 = M/12 units.
When the hands form an angle, the units between them is angle/6.
Arcs = M/12 Angle= Arc/6
M = 60H (+-) 2A outlet pipe can drain the same tank in 8 hours. If a half-full tank is first drained in two hours and is filled while the outlet pipe is still open, how long should the inlet pipe be opened to fill the tank?
M = no. Of minutes when the 2 hands form the specified angle
H = no. Of hours
A = angle the 2 hands form in the final position
Use + when the minute hand is “farther” than the hour hand in the final position
Use - when the hour hand is “farther” than the minute hand in the final position
When the hands point in the same direction, then A = 0
At what time after 12:00 noon will the hour hand and minute hand of a clock first form an angle of 120 degrees?
H = 0 (since the hour starts at 12 noon)
A = 120
We use + here since the minute-hand “overtakes” the hour hand in the final position
M = 60(0) + 2(12)
M = 21.818
The hands of the clock will first form 120 degrees the first time at 12:21.818 PM