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Algebra II
Worded Algebra Problems

I
...
Age Problems
Miscellaneous:
If X is half the age of Y m years ago:

y m
x= −
2 2
If X is double the age of Y in m years:
x = 2 y + 2m

Modifiers of time :
Decade = 10 years
Score = 20 years
Silver = 25 years
Golden = 50 years
Diamond = 75 years
Century or Centennial = 100 years

Other:
No
...
of years to come = Future Age – Present Age

Example 1:
The sum of Kim’s and Kevin’s ages is 18
...
4 years ago,
their ages were in the ratio 8:11
...


4 x − 4 : 5 x − 4 = 8 : 11
44 x − 44 = 40 x − 32
4 x = 12
x=3
4(3) = 12

Example 3:
On a man’s tombstone, it is said that one sixth of his life was spent in childhood
and one twelfth as a teenager
...

Alas, the son died four years before he did
...
How old did the man live to be?
x x x
x = + + +5+ y + 4
6 12 7
x = 2y
25
x=
x+9
28
3
x=9
28
x = 84

II
...
The sum of the digits is 14
...

u
h = 2t =
2
h + t + u = 14
h
h + + 2h = 14
2
h=4
4
t= =2
2
u = 2(4) = 8
100h + 10t + u = 100(4) + 10(2) + 8 = 428

Example 2:
The sum of digits of two digit number is 12
...
Find the original number
...
If 396 be
subtracted from the number, the order of the digits will be reversed
...

h = 2u → 1st
100h + 10t + u − 396 = 100u + 10t + h → 2nd
99h − 99u = 396
h − u = 4 → 2nd
h + t + u = 17 → 3rd
1st  2nd
2u − u = 4
u=4
h=8

t = 17 − 8 − 4
t =5
100(8) + 10(5) + 4 = 854

III
...
Minute hand method
x
m= xn
12

where x is the number of minute spaces the

hand will rotate
...
In 4 minutes, the hour hand will be
directly opposite the position occupied by the minute hand 3 minutes ago
...
3636

Example 2:
The time is past 2 pm
...
What time is it?

 Before =  After
60 H − 11M 60 H − 11( M + 10)
=
−2
2
60(2) − 11M 60(2) − 11( M + 10)
=
−2
2
M = 5
...


2
...


3
...

M travels
60 min
A = 6M
let M be the time in minutes; The hour hand travels
x1
30 
=
M 60 min 30 degrees in 60 minutes, every time the minute
hand makes a revolution
...
5M

x2 720 
=
H 24hrs

let H be the time in hours; the hour hand travels 720
degrees in 24 hours
...
5M

 = A− B
A = 6M
B = 30 H + 0
...
5M )

Final Formula:

60 H  2
M=
11

where H is the starting/origin hour
where Θ is the angle of the hour hand
between the minute hand
where M is the minutes passed

4
...


Shift + Mod -> Stat -> Linear Regression ( y = A+ Bx)
where x will be the minute hand’s number of dials
X
Y
travelled
where y will be the hour hand’s number of dials
0
15
travelled
Take 3:00 pm as a reference point
60
-40
At the beginning, the minute hand travelled no dials,
while the hour hand advances by 15 dials (15 minutes)
After 1 hour ( 4:00 pm) the minute hand is now ahead
of 60 dials, while the hour hand is left behind by 40 dials = - 40

ii
...
Number Problems
Equation :

Amx  Bn = C

then

where x is the unknown number
a, b, and c are constants
the signs are determined when more
than(+) and less than (-)
m, and n are multipliers
if half then ½, if double then 2, if 75%
0
...


Miscellaneous
◈ Consecutive Integers
n, n + 1, n + 2, n + 3
◈ Consecutive Even Integers
n, n + 2, n + 4, n + 6
where n is an even number
◈ Consecutive Odd Integers
n, n + 2, n + 4, n + 6
where n is an odd number

V
...
Area Formulas
*Mostly Geometric/Solid
(bh, s^2)
Mensuration Formulas*
2
...
Surface Area Formulas
[ 2(lw + wh + hl), 6s]
4
...
Motion Problems
Type 1 : Land Vehicle Motion
Equation:
Case 1 : Opposite Direction

D
v=
t

D1 + D2 =  D
Case 2 : Same Direction

D1 = D2

Type 2 : Water Vehicle Motion
Case 1 : Upstream
Upstream Speed = Boat Speed – Stream Speed
Case 2 : Downstream
Downstream Speed = Boat Speed + Stream Speed

Also
Speed of Boat in Still Water =

Speed of Stream =

d −u
2

Where d = downstream speed
u = upstream speed

d +u
2

Type 3 : Air Vehicle Problems
Case 1 : Headwind
HS = Speed of Plane – Speed of Wind
Case 2 : Tailwind
TS = Speed of Plane + Speed of Wind

VII
...
Five men were working at the start and after 10
days 5 men were added
...

Where Tabc is the time total time Person/Object A/B/C
can complete a certain task
A
B
C
ABC
Use + if the object is given to contribute to the
assigned and wanted task to be performed
Use – if the object is given to detriment the assigned
and wanted task to be performed(e
...
Pipes)

1
1
1
1
 
=
T T T
T

Example
A and B working together can finish painting a home in 6 days
...
How long will it take each of them
to finish the work alone?

1 1 1
+ =
A B 6
A = B −5
1
1 1
+ =
B −5 B 6
2
B − 17 B + 30 = 0
B = 15

Case 3 : Unity Work
1 1 1
1 1 1
 + + (t A ) +  + + (t B ) +
...
Working together they can roof a house in 5 days
...
Mixture Problems
Case 1 : Simple Mixture
Equations:
Ax + By = Cz
A+B=C
Where A,B, C are quantities of mass
Where x,y, z are percentages of those mass in the mixture

Case 2 : Mixture and Alligation
Allegation is a rule that enables us to find the ratio in which two ore more ingredients at the
given price must be mixed to produce a mixture of a desired price
...

Note : The mean price is NOT equivalent to
(expensive price + cheap price)/2
...
Always assume C:E
...
From A-B = 2

Example:
In what ratio must a shopkeeper mix two type of rice worth Rs
...
58/- per kg
so that by selling the mixture at Rs
...
30/- per kg and he may gain 10 percent?
Selling price of 1 kg of the mixture is 58
...
3/110%)
= 53
...
Suppose, a container contains x units
of a liquid from which y units are taken out and replaced by quantity (water is commonly
used)
...



x 1 −


y

x

n

Example
A container contains 50 litres of milk
...
This process is repeated one more time
...

Where Σc is the summation of change in the old basis solution
(Assume that the added substance/solution to the old/basis
solution is not the same as the old/basis solution)
Where ΣΔA/B is summation of change in the objective
substance after mixture
Where ΣΔc is the total summation of change in the whole
solution
...
What is the percent concentration in the
mixture?
0
...
35(15)+0
...
55%

Example 2:
How many pounds of lima beans that cost 0
...
50
php per pound to make a mixture of vegetables that costs
0
...
9x+0
...
65(x+16)
x = 9
...
If 2 kilograms of water evaporate
from 8 kilograms of solutions Y and 2 kilograms of solution Y are added to the remaining 6
kilograms of liquid, what percent of this new liquid solution is liquid X?
Find: x’
Given data:
Equation:
x = 0
...
30(8) + 0
...
375
C +  C
8 + (−2 + 2)

Therefore x’=37
...
Monetary Problems
Case 1 : Cost and Investment
Capital + Profit = Selling Price + Discount
Profit = x% of Capital
Discount = y% of Selling Price

Case 2 : Basic Interest
1
...
Compound Interest

F = P(1 + i )

n

;where I = Interest, P = Principality, i = interest rate, n = time,
F = Future Worth

Case 3 : Mixture-Related Problems
Equations:
Ax + By = Cz
A+B=C
Where A,B, C are quantities of money
Where x,y, z are investment percentages

Example 1:
A bookstore purchased a best selling book at 200 php per copy
...
3C=S-0
...
3(200)=0
...
Rudy then sold the VCR to Noel at a
profit of 20%
...
4x=selling price to Rudy
1
...
4x)=2856+x
X=4200 php

Example 3:
By selling balut at 5 php per dozen, a vendor gains 20%
...
5%
...

C + 0
...
6875)-1=0
...
2C=5 php
%= 6
...
125(25/6)=4
...
Diophantine Equations
-They are the equations where the known number of equations is less or insufficient to the
number of unknowns; where one unknown is going to be assumed/tried by any numbered
value/constant
Example : A merchant has three items on sale, namely a radio for 50 php, a clock for 30 php,
and a flashlight for 1 php
...
How many radios did he sale?
let x = radio, y = clock, z = flashlight
...
The denominator of a certain fraction is two more
than thrice the numerator
...

Find the original fraction
...
5
N
5
...
5) + 2 37

1
11
1
2
3
4
5
6
7
8
9
4
3
2
0
5
2
...
If the digits are
reversed in order and the resulting number
subtracted from the original number, the
difference is also 27
...
In a certain community of 1200 people, 60 percent
are toxic
...
What is the male
population?

Time’s Up!

Answer : 800 people
M+F = 1200
0
...
60)F=0
...
70M+(0
...
60(1200)
M = 800
...
Oden family has two daughters, Marina and Airi,
the sum of their ages are 18
...
In how many
years will Airi be 18 years old?

Time’s Up!

Answer : 13 years
Sum of their ages:
M + A = 18 1st equation
After three years
M+3=2(A+3) 2nd equation
Combining 1st and 2nd will gain
M = 13 yrs old
A = 5 yrs old
Airi will be 18 in (18-5) = 13 years

1
11
1
2
3
4
5
6
7
8
9
4
3
2
0
5
5
...
00027
1 second (1 hr / 3600 seconds)
=2
...
A sum of money is to be distributed among A, B,
C, and D in the proportion 5:2:4:3
...
Two cubes of sides 3m, 4m, and 5m are melted to
form a new cube
...


Time’s Up!

Answer : 6
...
0000m
3

3

V = s

3

'3

216 = s

'3

216 = s
6
...
What time after 1:00 will the hands of the clock be
together for the first time?

Time’s Up!

Answer : 1:05
...
Lita has ten bills in her wallet
...
If she has one more php 5 bill than php
10 bills and two more php 1 bills than php 5 bills,
how many php 10 bills does she have?

Time’s Up!

Answer : 12
...
Find 3 consecutive positive integers such that
when 5 times the largest be subtracted from the
square of the middle one, the result exceeds three
times the smallest by 7
...
A car covers a certain distance in three
hours at the speed of 124 kph
...
5567 fps
d
v=
t
d car = d truck + 120
(124kph)(3hrs ) = d truck + 120km
252km = d truck
vtruck

252km
=
= 84kph  76
...
A man can row at a speed of 6 kph in still
water
...
Find the
current rate

2
1
3
11
1
2
3
4
5
6
7
8
9
1
9
8
7
6
5
4
3
2
0

Answer : 2
...
Eleven samurai could kill the shogun in
14 days
...
How many days will it take
them to finish the job?

2
1
3
11
1
2
3
4
5
6
7
8
9
1
9
8
7
6
5
4
3
2
0

Answer : 18
...
2222 days

Time’s Up!

4
...
If my father is 6 years older than my
mother, find my present age
...
A cow is tethered in the middle of a field
with a 14 feet long rope
...


2
1
3
11
1
2
3
4
5
6
7
8
9
1
9
8
7
6
5
4
3
2
0

Answer : 12
...
3150days
50

Time’s Up!

6
...
What is the percent
concentration of sugar in the resulting
mixture?

2
1
3
11
1
2
3
4
5
6
7
8
9
1
9
8
7
6
5
4
3
2
0

Answer : 44 percent

0
...
44
40 + 10
'

Time’s Up!

7
...
What is
the length of the piece to be cut off?

2
1
3
11
1
2
3
4
5
6
7
8
9
1
9
8
7
6
5
4
3
2
0

Answer : 0
...
4142(rejected )
x+

2-2x

+

x

x2 = 0
...
A chemical plant has four tanks (A, B, C, and D) each
containing 1000 litres of a chemical is being pumped from
one tank to another as follows: (Units are in litres/min)
1
...
From C-A = 90
3
...
From C-D= 50
5
...
From D-B=110
Which tank will be emptied first?

5
4
3
2
1
6
11
1
2
3
4
5
6
7
8
9
1
9
8
7
6
5
4
3
2
0

Answer : Tank D
For Tank A:

For Tank C:

1000 L
1000 L
= 16
...
67 min
For Tank B:

Tank C’s volume will empty in 25 min
For Tank D:

1000 L
1000 L
= 33
...
33 min

Tank D’s volume will empty in 20 min

Difficult

Time’s Up!

2
...
Rick orders 10 ramen, 4 plates of
meat, and 1 cup of coffee, and paid 80 usd
...
Daryl ordered 5 each
...
0000 USD
1st equation : 10x + 4y + z = 80
2nd equation : 7x + 3y + z = 60
3rd equation : 5x + 5y + 5z = ???
Eliminating z from 1st and 2nd will yield
3x+y=20
(4th equation)
Eliminating x from 1st and 2nd will yield
-2y-3z=-40
2y+3z =40
(5th equation)
Adding 4th and 5th equation will yield
3x + 3y + 3z = 60
x+y+z=20
5x+5y+5z=100

Daryl paid 100 USD overall

Difficult

Time’s Up!

3
...

They realize that the article is difficult to understand
and they would require team work to finish the
assignment
...
If they divide the article into 3 parts
to that all of three of them can spend equal amount
of time on the article, the number of pages B should
read is?

5
4
3
2
1
6
11
1
2
3
4
5
6
7
8
9
1
9
8
7
6
5
4
3
2
0

Answer : 24 pages
1 1 1
x + +  = 78
2 3 4
13 pages
(
) x = 78 pages
12 min
x = 72 min
_______
72 min
B=
= 24 pages
3 min
page

5
4
3
2
1
6
11
1
2
3
4
5
6
7
8
9
1
9
8
7
6
5
4
3
2
0

Difficult
4
...
Running in opposite
directions, they meet every 10 seconds
...
Find the rates in fps

Time’s Up!

Answer : 18 fps and 12 fps
For opposite direction
(1st equation)

10 A + 10 B = 300
A + B = 30

For same direction
(2nd equation)

Combining 1st and 2nd equation will yield
A = 18 fps
B = 12 fps

50 A = 50 B + 300
A− B = 6

5
4
3
2
1
6
11
1
2
3
4
5
6
7
8
9
1
9
8
7
6
5
4
3
2
0

Difficult
5
...
10 men and 21
women can finish the same work in 3 days
...
0000 days
Work Problem Case 1

(12men)(6days ) = 72man − day
10men
21women
1
+
=
72man − day xwomen − day 3days
x = 108women − day
108women − day
= 9

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