Unit 1
Fractions, Ratios & Proportions
Six is four more than \(\dfrac{2}{3}\) of what number?
β‘ Memory Point
"IS / OF = %/100" β "is" β equals, "of" β multiply.
Translate: 6 = (2/3)Β·x + 4. Isolate x step by step.
Translate: 6 = (2/3)Β·x + 4. Isolate x step by step.
ν΄μ€ (Korean Explanation)
λ¬Έμ λ₯Ό μμΌλ‘ λ²μν©λλ€: 6 = (2/3)x + 4
μλ³μμ 4λ₯Ό λΉΌλ©΄: 2 = (2/3)x
μλ³μ 3/2λ₯Ό κ³±νλ©΄: x = 2 Γ (3/2) = 3
μλ³μμ 4λ₯Ό λΉΌλ©΄: 2 = (2/3)x
μλ³μ 3/2λ₯Ό κ³±νλ©΄: x = 2 Γ (3/2) = 3
6 = (2/3)x + 4 β (2/3)x = 2 β x = 3 β
A stock clerk had 600 pads on hand. He then issued \(\dfrac{3}{8}\) to Division X, \(\dfrac{1}{4}\) to Division Y, and \(\dfrac{1}{6}\) to Division Z. How many pads remain?
β‘ Memory Point β LCD Method
LCD (Least Common Denominator): Find LCD of 8, 4, 6 = 24.
Add fractions with same denominator, then subtract from 1.
Add fractions with same denominator, then subtract from 1.
ν΄μ€
LCD = 24λ‘ ν΅λΆν©λλ€:
\(\frac{3}{8} = \frac{9}{24}\), \(\frac{1}{4} = \frac{6}{24}\), \(\frac{1}{6} = \frac{4}{24}\)
ν©κ³: \(\frac{9+6+4}{24} = \frac{19}{24}\) μ§κΈλ¨
λ¨μ λΉμ¨: \(1 - \frac{19}{24} = \frac{5}{24}\)
λ¨μ ν¨λ: \(600 \times \frac{5}{24} = \mathbf{125}\)
\(\frac{3}{8} = \frac{9}{24}\), \(\frac{1}{4} = \frac{6}{24}\), \(\frac{1}{6} = \frac{4}{24}\)
ν©κ³: \(\frac{9+6+4}{24} = \frac{19}{24}\) μ§κΈλ¨
λ¨μ λΉμ¨: \(1 - \frac{19}{24} = \frac{5}{24}\)
λ¨μ ν¨λ: \(600 \times \frac{5}{24} = \mathbf{125}\)
600 Γ (5/24) = 125 pads β
The winner of a race received \(\dfrac{1}{3}\) of the total purse. The third-place finisher received one-third of the winner's share. If the winner's share was $2,700, what was the total purse?
β‘ Memory Point β INVERSE Fraction
"Part = Fraction Γ Whole" β "Whole = Part Γ· Fraction"
Winner = (1/3) Γ Total β Total = Winner Γ 3
Winner = (1/3) Γ Total β Total = Winner Γ 3
ν΄μ€
μ°μΉμ λͺ« = μ 체 μκΈμ 1/3 = $2,700
λ°λΌμ: μ 체 μκΈ = $2,700 Γ 3 = $8,100
(3μ μ μμ λͺ« = $2,700 Γ 1/3 = $900μ ν¨μ β λ¬Έμ μμ 묻λ κ²μ μ 체 μκΈ!)
λ°λΌμ: μ 체 μκΈ = $2,700 Γ 3 = $8,100
(3μ μ μμ λͺ« = $2,700 Γ 1/3 = $900μ ν¨μ β λ¬Έμ μμ 묻λ κ²μ μ 체 μκΈ!)
Total = $2,700 Γ· (1/3) = $2,700 Γ 3 = $8,100 β
A box contains red and blue marbles in the ratio 3 : 5. If there are 160 marbles in total, how many blue marbles are there?
β‘ Memory Point β RATIO UNIT
1 unit = Total Γ· Sum of parts
3:5 β 8 units total β 1 unit = 160 Γ· 8 = 20
Blue = 5 Γ 20
3:5 β 8 units total β 1 unit = 160 Γ· 8 = 20
Blue = 5 Γ 20
ν΄μ€
λΉμ¨μ ν© = 3 + 5 = 8 λ¨μ
1 λ¨μ = 160 Γ· 8 = 20κ°
νλ κ΅¬μ¬ = 5 Γ 20 = 100κ°
1 λ¨μ = 160 Γ· 8 = 20κ°
νλ κ΅¬μ¬ = 5 Γ 20 = 100κ°
Blue = (5/8) Γ 160 = 100 β
Unit 2
Percentages & Discounts
What is 30% of 150?
β‘ Memory Point β PERCENT Γ WHOLE
"%" means "per hundred" β 30% = 30/100 = 0.30
Move decimal: 150 Γ 0.3 = 15 Γ 3 (mental math trick!)
Move decimal: 150 Γ 0.3 = 15 Γ 3 (mental math trick!)
ν΄μ€
30% = 0.30
150 Γ 0.30 = 45
150 Γ 0.30 = 45
150 Γ (30/100) = 150 Γ 0.3 = 45 β
A jacket originally costs $120. It is on sale at a 25% discount. What is the sale price?
β‘ Memory Point β KEEP % TRICK
Discount 25% β Keep 75%
Sale Price = Original Γ (1 β discount%)
Faster: 120 Γ 0.75 (one step!)
Sale Price = Original Γ (1 β discount%)
Faster: 120 Γ 0.75 (one step!)
ν΄μ€
ν μΈμ¨ 25% β λ΄κ° λ΄λ κΈμ‘ = 75%
$120 Γ 0.75 = $90
$120 Γ 0.75 = $90
$120 β ($120 Γ 0.25) = $120 β $30 = $90 β
A price increased from $80 to $100. What is the percent increase?
β‘ Memory Point β % CHANGE Formula
% Change = (New β Old) / Old Γ 100
β οΈ Always divide by the ORIGINAL (Old) value, not the new one!
β οΈ Always divide by the ORIGINAL (Old) value, not the new one!
ν΄μ€
μ¦κ°λ = $100 β $80 = $20
% μ¦κ° = $20 Γ· $80 Γ 100 = 25%
β οΈ ν¨μ : $20 Γ· $100 = 20% (νλ¦Ό! μλ κ°μΌλ‘ λλ μΌ ν¨)
% μ¦κ° = $20 Γ· $80 Γ 100 = 25%
β οΈ ν¨μ : $20 Γ· $100 = 20% (νλ¦Ό! μλ κ°μΌλ‘ λλ μΌ ν¨)
% increase = (100 β 80) / 80 Γ 100 = 25% β
Unit 3
Averages & Statistics
A student scored 72, 85, 90, and 68 on four tests. What is the average score?
β‘ Memory Point β AVERAGE
Average = Sum Γ· Count
Quick add trick: pair numbers close to 80 β (72+88) noβ¦ just add left to right carefully.
Quick add trick: pair numbers close to 80 β (72+88) noβ¦ just add left to right carefully.
ν΄μ€
ν©κ³: 72 + 85 + 90 + 68 = 315
νκ· : 315 Γ· 4 = 78.75
νκ· : 315 Γ· 4 = 78.75
(72 + 85 + 90 + 68) Γ· 4 = 315 Γ· 4 = 78.75 β
The average of five numbers is 20. If four of the numbers are 14, 18, 22, and 26, what is the fifth number?
β‘ Memory Point β REVERSE AVERAGE
Sum = Average Γ Count (reverse of average formula)
Then: Missing = Total Sum β Known Sum
Then: Missing = Total Sum β Known Sum
ν΄μ€
μ 체 ν© = νκ· Γ κ°μ = 20 Γ 5 = 100
λ€ μμ ν©: 14 + 18 + 22 + 26 = 80
λ€μ― λ²μ§Έ μ = 100 β 80 = 20
λ€ μμ ν©: 14 + 18 + 22 + 26 = 80
λ€μ― λ²μ§Έ μ = 100 β 80 = 20
5th number = (20 Γ 5) β (14+18+22+26) = 100 β 80 = 20 β
Unit 4
Distance, Rate & Time
A train travels at 60 mph for 2.5 hours. How far does it travel?
β‘ Memory Point β DRT Triangle
D = R Γ T | R = D/T | T = D/R
Draw a triangle: D on top, RΒ·T on bottom. Cover what you need!
Draw a triangle: D on top, RΒ·T on bottom. Cover what you need!
ν΄μ€
D = R Γ T = 60 Γ 2.5 = 150 miles
Distance = 60 mph Γ 2.5 hr = 150 miles β
Two cars start toward each other along a straight road between two cities that are 450 miles apart. The first car travels at 35 mph, and the second at 48 mph. How long will it take them to meet?
β‘ Memory Point β CLOSING SPEED
Two objects moving TOWARD each other β Add speeds
Combined rate = 35 + 48 = 83 mph
Time = Distance Γ· Combined Rate
Combined rate = 35 + 48 = 83 mph
Time = Distance Γ· Combined Rate
ν΄μ€
ν©μ° μλ = 35 + 48 = 83 mph
λ§λλ μκ° = 450 Γ· 83 β 5.42 μκ°
λ§λλ μκ° = 450 Γ· 83 β 5.42 μκ°
T = 450 Γ· (35 + 48) = 450 Γ· 83 β 5.42 hrs β
Unit 5
Work Rate Problems
Two workers can complete a job β one in 6 hours, the other in 3 hours. Working together, how long will they take?
β‘ Memory Point β WORK RATE
Rate = 1/Time done alone
Combined rate = sum of individual rates
\(\frac{1}{T} = \frac{1}{A} + \frac{1}{B}\)
Combined rate = sum of individual rates
\(\frac{1}{T} = \frac{1}{A} + \frac{1}{B}\)
ν΄μ€
1μκ°λΉ μμ
λ₯ : 첫 λ²μ§Έ = 1/6, λ λ²μ§Έ = 1/3 = 2/6
ν©μ° = 1/6 + 2/6 = 3/6 = 1/2 (1μκ°λΉ)
ν¨κ» 걸리λ μκ° = 1 Γ· (1/2) = 2μκ°
ν©μ° = 1/6 + 2/6 = 3/6 = 1/2 (1μκ°λΉ)
ν¨κ» 걸리λ μκ° = 1 Γ· (1/2) = 2μκ°
1/T = 1/6 + 1/3 = 1/2 β T = 2 hours β
One man can load a truck in 25 min, a second in 50 min, and a third in 10 min. How long would it take all three working together?
β‘ Memory Point β THREE WORKERS
Add all three rates: 1/25 + 1/50 + 1/10
Find LCD = 50. Convert each: 2/50 + 1/50 + 5/50 = 8/50
Time = 50/8 = 6.25 = \(6\frac{1}{4}\)
Find LCD = 50. Convert each: 2/50 + 1/50 + 5/50 = 8/50
Time = 50/8 = 6.25 = \(6\frac{1}{4}\)
ν΄μ€
LCD = 50μΌλ‘ ν΅λΆ:
\(\frac{1}{25} = \frac{2}{50}\), \(\frac{1}{50} = \frac{1}{50}\), \(\frac{1}{10} = \frac{5}{50}\)
ν©μ° = \(\frac{2+1+5}{50} = \frac{8}{50} = \frac{4}{25}\)
걸리λ μκ° = \(\frac{25}{4} = 6\frac{1}{4}\) λΆ
\(\frac{1}{25} = \frac{2}{50}\), \(\frac{1}{50} = \frac{1}{50}\), \(\frac{1}{10} = \frac{5}{50}\)
ν©μ° = \(\frac{2+1+5}{50} = \frac{8}{50} = \frac{4}{25}\)
걸리λ μκ° = \(\frac{25}{4} = 6\frac{1}{4}\) λΆ
T = 1 Γ· (8/50) = 50/8 = 6.25 = 6ΒΌ min β
Unit 6
Algebra & Equations
If \(4x - y = 20\) and \(2x + y = 28\), then \(x =\) ?
β‘ Memory Point β ELIMINATION
ADD equations when signs are opposite (βy and +y cancel!)
4x β y + 2x + y = 20 + 28 β 6x = 48
4x β y + 2x + y = 20 + 28 β 6x = 48
ν΄μ€
λ μμ λν©λλ€: (4x β y) + (2x + y) = 20 + 28
6x = 48 β x = 8
κ²μ¦: 4(8) β y = 20 β y = 12 | 2(8) + 12 = 28 β
6x = 48 β x = 8
κ²μ¦: 4(8) β y = 20 β y = 12 | 2(8) + 12 = 28 β
Add equations: 6x = 48 β x = 8 β
If \(6 + x + y = 20\) and \(x + y = k\), then \(20 - k =\) ?
β‘ Memory Point β SUBSTITUTION SHORTCUT
Don't solve for x and y separately!
From the first equation: 6 + k = 20, so k = 14
Then 20 β k = ?
From the first equation: 6 + k = 20, so k = 14
Then 20 β k = ?
ν΄μ€
x + y = kλ₯Ό 첫 λ²μ§Έ μμ λμ
:
6 + k = 20 β k = 14
λ°λΌμ: 20 β k = 20 β 14 = 6
6 + k = 20 β k = 14
λ°λΌμ: 20 β k = 20 β 14 = 6
6 + k = 20 β k = 14 β 20 β k = 6 β
Solve for \(x\): \(3x + 7 = 22\)
β‘ Memory Point β ISOLATE x
UNDO in reverse order: Addition first, then multiplication
Step 1: subtract 7. Step 2: divide by 3.
Step 1: subtract 7. Step 2: divide by 3.
ν΄μ€
3x + 7 = 22
3x = 22 β 7 = 15
x = 15 Γ· 3 = 5
3x = 22 β 7 = 15
x = 15 Γ· 3 = 5
3x = 15 β x = 5 β
Unit 7
Geometry, Area & Interest
A rectangle has a length of 12 and a width of 5. What is its perimeter?
β‘ Memory Point β P vs A
Perimeter = 2(l + w) β add length + width, then double
Area = l Γ w β multiply (don't confuse them!)
Area = l Γ w β multiply (don't confuse them!)
ν΄μ€
λλ = 2 Γ (κΈΈμ΄ + λλΉ) = 2 Γ (12 + 5) = 2 Γ 17 = 34
β οΈ λμ΄(60)μ νΌλ μ£Όμ! λμ΄ = 12 Γ 5 = 60
β οΈ λμ΄(60)μ νΌλ μ£Όμ! λμ΄ = 12 Γ 5 = 60
P = 2(12 + 5) = 2 Γ 17 = 34 β
A triangle has a base of 8 cm and a height of 6 cm. What is its area?
β‘ Memory Point β HALF BASE HEIGHT
Area of triangle = Β½ Γ b Γ h
A triangle is HALF of a rectangle β always divide by 2!
A triangle is HALF of a rectangle β always divide by 2!
ν΄μ€
λμ΄ = Β½ Γ λ°λ³ Γ λμ΄ = Β½ Γ 8 Γ 6 = 24 cmΒ²
β οΈ ν¨μ : 8 Γ 6 = 48 (2λ‘ λλλ κ²μ μμΌλ©΄ νλ¦Ό!)
β οΈ ν¨μ : 8 Γ 6 = 48 (2λ‘ λλλ κ²μ μμΌλ©΄ νλ¦Ό!)
A = Β½ Γ 8 Γ 6 = 24 cmΒ² β
A principal of $500 is invested at 4% simple interest per year. How much interest is earned in 3 years?
β‘ Memory Point β SIMPLE INTEREST
I = P Γ R Γ T (Principal Γ Rate Γ Time)
β οΈ Simple interest β compound interest. Interest stays the same each year.
β οΈ Simple interest β compound interest. Interest stays the same each year.
ν΄μ€
I = P Γ R Γ T = $500 Γ 0.04 Γ 3 = $60
β οΈ ν¨μ : $560μ μκΈ+μ΄μ ν©κ³, $620μ μ€κ³μ° β λ¬Έμ λ μ΄μλ§!
β οΈ ν¨μ : $560μ μκΈ+μ΄μ ν©κ³, $620μ μ€κ³μ° β λ¬Έμ λ μ΄μλ§!
I = $500 Γ 0.04 Γ 3 = $60 β
Challenge: 20 liters of a 30% acid solution is mixed with 30 liters of a 50% acid solution. What is the concentration of the resulting mixture?
β‘ Memory Point β MIXTURE Formula
% mix = (VβΒ·Cβ + VβΒ·Cβ) / (Vβ + Vβ)
Think: total acid amount Γ· total volume
Think: total acid amount Γ· total volume
ν΄μ€
μ° μ±λΆ ν©κ³:
20L Γ 30% = 6L μ°
30L Γ 50% = 15L μ°
μ΄ μ° = 6 + 15 = 21L / μ΄ λΆνΌ = 50L
λλ = 21/50 = 0.42 = 42%
20L Γ 30% = 6L μ°
30L Γ 50% = 15L μ°
μ΄ μ° = 6 + 15 = 21L / μ΄ λΆνΌ = 50L
λλ = 21/50 = 0.42 = 42%
(20Γ0.30 + 30Γ0.50) / 50 = 21/50 = 42% β