Equalising Units Concept (Remainder)

Given: Before, One Side, Change Ratios, After Amount
Find: Change Amount

First
At first, the amount of money Abel had was 3/5 of the amount of money Simon had. They went for dinner together. Abel paid 1/4 of the bill and Simon paid for the rest. After paying for the dinner, Abel has $63 left and Simon has 1/4 of his money left. How much was the dinner?

(Before)
Abel : Simon
= 3 : 5
= 12 : 20

(Simon)
After : Change : Before
= 1 : 3 : 4
= 5 : 15 : 20

(Change)
Abel : Simon : Total
= 1 : 3 : 4
= 5 : 15 : 20

(After)
Abel = 12 u - 5 u = 7 u
7 u = $63

(Change)
Total = 20 u
20 u = $180

Equalising Units Concept (Unchanged Side)

First
There were 200 supporters for a school's soccer team during a match. 40% of them were girls. Later, more girls joined the group of supporters during half-time. The percentage of the girls increased to 70%. Find the number of girls who joined the group of supporters during half-time.

Equalising Unchanged Sides Method

(Before)
Girls : Boys
= 40% : 60%
= 2 : 3

(After)
Girls : Boys
= 70% : 30%
= 7 : 3

(Before)
Total = 2 u + 3 u = 5 u
5 u = 200

(Change)
Girls = 7 u - 2 u = 5 u = 200

Finding Unchanged Amount Method

(Before)
Girls = 40% x 200 = 80 girls
Boys = 60% x 200 = 120 boys

(After)
Boys (30%) = 120 boys
Girls (70%) = 120 / 30% x 70% = 280 girls

(Change)
Girls = 280 girls - 80 girls = 200 girls

Second
There were total of 720 pupils at a stadium. 30% of the pupils were girls. When more giris came into the stadium, the percentage of girls increased to 40%. How many more girls came into the stadium?

Equalising Units Method

(Before)
Girls : Boys
= 30% : 70%
= 3 : 7
= 9 : 21

(After)
Girls : Boys
= 40% : 60%
= 2 : 3
= 14 : 21

(Before)
Total = 9 u + 21 u = 30 u
30 u = 720

(Change)
Girls = 14 u - 9 u = 5 u
5 u = 720 / 6 = 120

Finding Amount Method

(Before)
Girls = 30% x 720 = 216 girls
Boys = 70% x 720 = 504 boys

(After)
Boys (60%) = 504 boys
Girls (40%) = 504 / 60% x 40% = 336 girls

(Change)
Girls = 336 - 216 = 120 girls

Third
Clifford had a total of 600 Malaysia and Singapore stamps. 48% of his stamps were Malaysia stamps. When he collected more Malaysia stamps, the percentage of Singapore stamps decreased to 25%. How many new Malaysia stamps had he collected?

Equalising Units Method

(Before)
Malaysia : Singapore
= 48% : 52%
= 12 : 13

(After)
Malaysia : Singapore
= 75% : 25%
= 3 : 1
= 39 : 13

(Before)
Total = 12 u + 13 u = 25 u
25 u = 600
1 u = 24

(Change)
Malaysia = 39 u - 12 u = 27 u
27 u = 648

Finding Amount Method

(Before)
Malaysia = 48% x 600 = 288
Singapore = 52% x 600 = 312

(After)
Singapore (25%) = 312
Malaysia (75%) = 312 x 3 = 936

(Change)
Malaysia = 936 - 288 = 648

Fourth
Matthew had a total of 450 Japanese and Korean comics. 36% of them were Japanese comics. When he bought more Japanese comics, the percentage of Korean comics decreased to 48%. How many new Japanese comics did Matthew buy?

Equalising Units Method

(Before)
Japanese : Korean
= 36% : 64%
= 9 : 16
= 27 : 48

(After)
Japanese : Korean
= 52% : 48%
= 13 : 12
= 52 : 48

(Before)
Total = 27 u + 48 u = 75 u
75 u  = 450 comics

(Change)
Japanese = 52 - 27 = 25 u
25 u = 450 / 3 = 150 comics

Gap Unit Concept (Headstart + Catchup)

First
Betty starts saving $30 per month from January. Dave starts saving $45 per month from March. In which month will Betty and Dave have an equal amount of savings?

(Headstart)
Gap Total = Betty = $30 per month x 2 months earlier = $60

(Catchup)
Gap Unit = Dave - Betty = $45 - $30 = $15 per month
Gap Number = $60 / $15 per month = 4 months

Second
Joseph reads 12 pages of a novel each day. His sister reads only 8 pages of the some novel each day but has started reading it 6 days earlier than Joseph. If Joseph began reading on 1st of February, on which date would his sister and he have read the same number of pages?

(Headstart)
Gap Total = Sister = 8 pages each day x 6 days earlier = 48 pages

(Catchup)
Gap Unit = Joseph - Sister = 12 - 8 = 4 pages each day
Gap Number = 48 pages / 4 pages each day = 12 days

1st of February + 12 days - 1 day = 12th of February (ordered range concept)

Third
Machine A prints 18 books per hour. Machine B prints 12 books per hour. How many hours must Machine A operate so that it will print the same number of books as Machine B given that Machine B has started operating 2 hours earlier than Machine A?

(Headstart)
Gap Total = Machine B = 12 books per hour x 2 hours earlier = 24 books

(Catchup)
Gap Unit = Machine A - Machine B = 18 - 12 = 6 books per hour
Gap Number = 24 books / 6 books per hour = 4 hours

Fourth
Patricia and James are selling funfair tickets. Every day, Patricia can sell 20 tickets while James can sell 14 tickets. If James started selling 9 days earlier than Patricia, how many days will Patricia take to sell the same number of tickets as James?

(Headstart)
Gap Unit Total = James = 14 tickets each day x 9 days earlier = 126 tickets

(Catchup)
Gap Unit Value = Patricia - James = 20 - 14 = 6 tickets each day
Gap Unit Number = 126 tickets / 6 tickets each day = 21 days

Area Concepts (Triangles with Common Bases or Heights)

Given: Common Base, Combined Height
Find: Combined Area

First
The figure below, not drawn to scale, shows two shaded triangles in a rectangle of length 20 cm and breadth 16 cm. What fraction of the whole figure is shaded?

(Express your answer in the simplest form.)

 

Method 1: Find Areas

Shaded Area
= Combined Area of 2 Triangles
= 1/2 × Common Base x Combined Height
= 1/2 × 12 cm × 16 cm
= 96 cm^2

Area of Rectangle
= length x breadth
= 20 cm x 16 cm
= 320 cm^2

Fraction (Shaded Area / Whole Area)
= 96 / 320 = 3 / 10

Method 2: Formulas in Ratios

Combined Height = Breadth

Shaded Area : Whole Area
= 1/2 × Common Base × Combined Height : Length x Breadth
= 1/2 × Common Base : Length
= 1/2 × 12 : 20
= 3 : 10
= 3 / 10

Units and Parts (Unchanged Amount and From Equality)

First
Jim bought some chocolates and gave half of it to Ken. Ken bought some sweets and gave half of it to Jim. Jim ate 12 sweets and Ken ate 18 chocolates. The ratio of Jim’s sweets to chocolates became 1 : 7. The ratio of Ken’s sweets to chocolates became 1 : 4. How many sweets did Ken buy?

(After)
(Jim)
Sweets : Chocolates
= 1u : 7u

(Ken)
Sweets : Chocolates
= 1p : 4p

Reverse the Changes

(Before)
(Sweets)
Jim Sweets = Ken Sweets
1u + 12 sweets eaten = 1p
7u + 84 = 7p

(Chocolates)
Jim Chocolates = Ken Chocolates
7u = 4p + 18 chocolates eaten
7u - 18 = 4p

84 -(-18) = 7p - 4p
3p = 102

Ken Sweets (Before) = 1p = 34

Conversion (Units)

Time Conversion

Concepts: 
6 Times Tables

Big to Small

First (Single Unit)
Convert from hours to minutes

2 hours = 2 x 60 minutes = 120 minutes

5 hours = 5 x 60 minutes = 300 minutes


Convert from minutes to seconds

7 minutes = 7 x 60 seconds = 420 seconds

8 minutes = 8 x 60 seconds = 480 seconds


Second (Mixed Units)
Convert from hours and minutes to minutes

4 hours 5 minutes 
= 4 x 60 minutes + 5 minutes
= 240 minutes + 5 minutes
= 245 minutes

6 hours 24 minutes 
= 6 x 60 minutes + 24 minutes
= 360 minutes + 24 minutes
= 384 minutes

Convert from minutes and seconds to seconds

12 minutes 54 seconds 
= 12 x 60 seconds + 54 seconds
= 720 seconds + 54 seconds
= 774 seconds

14 minutes 28 seconds 
= 14 x 60 seconds + 28 seconds
= 840 seconds + 28 seconds
= 868 seconds

21 minutes 41 seconds
= 21 x 60 seconds + 41 seconds
= 1260 + 41 seconds
= 1301 seconds


Small to Big

First (Single Unit: Whole Numbers)

Convert from minutes to hours

360 minutes = (360 / 60) hours = 6 hours



480 minutes = (480 / 60) hours = 8 hours



540 minutes = (540 / 60) hours = 9 hours


Convert from seconds to minutes


600 seconds = (600 / 60) minutes = 10 minutes


720 seconds = (720 / 60) minutes = 12 minutes


840 seconds = (840 / 60) minutes = 14 minutes

Second (Mixed Units, Mixed Numbers)

366 minutes = (366 / 60) hours = 6 hours 6 minutes
or
= 6 6/60 hours = 6 1/10 hours

492 minutes = (492 / 60) = 8 hours 12 minutes
= 8 12/60 hours = 8 1/5 hours

Patterns (Tokens)

Patterns (Rods)

Angles (Parallelogram and Triangle)

In the figure below, not drawn to scale, ABCD is a parallelogram, DE and AE are straight lines. Find the ∠AED.

Label the point with a 90 angle as F.

Method 1
CFE = 90 (adjacent angles on straight line BFC)
BCE = 65 (corresponding angles)
AED = 180 - 90- 65 = 25 (angle sum of triangle CFE)

Method 2
AFB = 90 (adjacent angles on straight line BFC)
ABC = 65 (diagonally opposite angles of parallelogram ABCD)
AED = BAF = 180 - 90 - 65 = 25 (angle sum of triangle BAF, alternate angles)

Angles (Isosceles Triangle)

The figure below is not drawn to scale. Given that TP = SP, find ∠RTS

TRS = 40 + 24 = 64 (exterior angle = interior opposite angles of triangle TPR)

TSP = (180 - 24) / 2 = 78 (base angles of isosceles triangle TSP)

RTS = 180 - 78 - 64 = 38 (angle sum of triangle RTS)

TRP = 180 - 40 - 24 = 116 (angle sum of triangle TRP)

TSP = (180 - 24) / 2 = 78 (base angles of isosceles triangle TSP)

RTS = 116 - 78 = 38 (exterior angle = interior opposite angles of triangle RTS)

Angles (2 Isosceles Triangles)

First

The figure below is not drawn to scale. ABC is an isosceles triangle where BA = BC. Given that D is the midpoint of BC and AC is 1/2 of BA, find ∠ADB.

Method 1

∠DCB = (180 - 20) / 2 = 80 (base angles of isosceles triangle ABC)

∠ADC = (180 - 80) / 2 = 50 (base angles of isosceles triangle ACD)

∠ADB = 180 - 50 = 130 (adjacent angles on straight line BDC)

Angles (Isosceles Triangle and Parallelogram)

First

The following figure is not drawn to scale. Given that ABCD is a parallelogram and ∠ADE is an isosceles triangle, find ∠y.

ADC = 180 - 118 = 62 (co-interior angles of parallelogram ABCD)

Y = 62 / 2 (exterior angle = interior opposite angles, base angles of isosceles triangle ADE)

DAB = 118 (diagonally opposite angles of parallelogram ABCD)

y = (180 - 118) / 2 = 31 (co-interior angles of trapezium ABCE)

Second

Study the figure below and find ∠FCB.

Method 1

FCD = EFA = (180 - 84) / 2 = 48 (base angles of isosceles triangle EAF, corresponding angles)

DCB = 180 - 68 = 112 (co-interior angles of parallelogram ABCD)

FCB = 112 - 48 = 64

Method 2

CFB = (180 - 84) / 2 = 48 (base angles of isosceles triangle EAF, vertically opposite angles)

FBC = 68 (diagonally opposite angles)

FCB = 180 - 48 - 68 = 64 (angle sum of triangle FCB)