#### If (a + b)^2 = 25 and (a - b)^2 = 45, then a^2 + b^2 = ?

(a+b)^2 = a^2+2ab+b^2 (a-b)^2 = a^2-2ab+b^2 a^2+2ab+b^2 = 25 a^2-2ab+b^2 = 45 (ADD) ------------------------------- (2a^2 + 2b^2 = 70)/2 a^2 + b^2 = 35

#### A car travels 120 miles at a speed of 40 miles per hour and makes a return trip at a speed of 60 miles per hour. What is its average speed, in miles per hour, of the police car?

SOLUTION: Rate = distance/time time = distance/rate Total distance: 120 × 2 = 240miles Total time: 1st trip = 120/40 = 3hrs Return = 120/60 = 2hrs 3hrz + 2hrs = 5hrs Average = Total distance ÷ total time 240m ÷ 5hrs = 48mph

#### 3/4 ÷ 1/4 =

Division of fraction, remember to do reciprocal of the divider and change "÷" to "×" 3/4 ÷ 1/4 = 3/4 × 4/1 = 12/4 = 3

#### A jeepney travels at 40kph while moving but after accounting for stopping time, to let passengers on and off the coach, it averages a speed of 34kph. How many minutes does the jeepney stop for each hour?

In one hour without stopping, the bus would have travelled 40km. Once stopping is factored in, the bus actually travels 34km. So it travels 6km less far as a consequence. 40 - 34 = 6km Distance/Speed = Time 6km/40kmph = 0.15 hours 0.15 x 60 = 9 minutes � So the jeepney stops for 9min in every hour on average.

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#### A solution is made of water and pure orange. If 80% of the solution is water, how many liters of pure orange are in 25 liters of this solution?

Given: 80% water 100%-80%=20%pure orange Solution: 20% of 25L is pure orange 20% × 25 = pure orange 0.2 × 25 = 5

#### John decided to cycle to his friend’s house at a speed of 6kph and the journey took 30 minutes. How far did John cycle?

SOLUTION: Rate = distance/time Rate × time = distance Distance = 6kph × 30min Distance = 6kph × 0.5hr Distance = 3 km

#### What is the average of 7/8 and 3/4?

(7/8 + 3/4)/2 = (7/8 + 6/8)/2 = (13/8)/2 = 13/16 To be much clearer, let's rewrite it 13/8 ÷ 2 = apply RULE of dividing FRACTION 13/8 ÷ 2 = 13/8 × 1/2 = 13/16

#### 1/4 % is equivalent to?

Remember how to convert percentage to simple decimal form. We always divide ÷100%. 1/4 % is equivalent to? 1/4% ÷ 100% = 1/4%×1/100%=1/400=0.0025 ...or.... 1/4% ÷ 100% = 0.25% ÷ 100% = 0.0025

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#### A company makes a profit equal to 25% of its sales. The profit is shared equally among the 4 owners of the company. If the company generates sales of $2,000,000, how much money does each one of the owners get?

2,000,000 × 0.25= 500,000 500,000 ÷ 4 = 125,000

#### 40% of 5 is equal to

40% of 5 is equal to of ---> "×" 40% × 5 = is ---> "=" 0.4 × 5 = 2

#### The population of a certain town is 5,000.If it increases 5% each year, what will be the approximate population at the end of 10 years?

The population of a certain town is 5,000. ---> population is 5000 If it increases 5% each year, ---> EACH YEAR +5% Q: what will be the approximate population at the end of 10 years? Solution#1 5000 × 1.05^10 = 5000 × 1.62889 = 8,144.45 = 8144✔ Solution#2: List method: 1.)5000×1.05 = 5250 2.)5250×1.05 = 5,512.5 3.)5,512×1.05 = 5,787.6 4.)5,788×1.05 = 6077 5.)6077×1.05 = 6,380.85 6.)6,381×1.05 = 6700.05 7.)6700×1.05 = 7035 8.)7035×1.05 = 7,386.75 9.)7,387×1.05 = 7,756.35 10.)7,756×1.05 = 8,143.8 8,143.8 = 8,144

#### If -6/(a - 3) = 3/(a + 2), then a = ?

-6/(a - 3) = 3/(a + 2) -6(a + 2) = 3 (a - 3) -6a - 12 = 3a - 9 -6a - 3a = -9 + 12 -9a = 3 a = 3/-9 a = - 1/3

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#### 10-2+4×2-(10÷5×2) = ?

APPLY the RULE not the acronym.... 10-2+4×2-(10÷5×2) = 10-2+4×2-(2×2) = 10-2+4×2-(4) = \/ 10-2 + 8 - 4 = \/ 8 + 8 - 4 = \/ 16 - 4 = 12

#### 1/2 % of a number is 2/5 % of 250. What is the number?

Note: of --> "×" is --> "=" Let n be the number Covert percentage to decimals: 1/2% ÷ 100% = 1/2% × 1/100% = 1/200 = 0.005 2/5% ÷ 100% = 2/5% × 100% = 2/500= 0.004 Solution: 1/2 % of a number is 2/5 % of 250. 1/2 % × n = 2/5 % × 250 0.005 × n = 0.004 × 250 n = 1/0.005 n = 200

#### What is 1/6 of 96% in decimal form?

1/6 × 96% Convert % to decimal by moving 2 decimal places to the left 1/6 × 0.96 = 0.96/6 = 0.16

#### 1/2 % of 20 is ?

Remember how to convert percentage to simple decimal form. We always divide ÷100%. 1/2% ÷ 100% = 1/2%×1/100%=1/200 = 0.005 0.5% ÷ 100% = 0.005 1/2 % of 20 is ? \/ 0.5% × 20 = \/ 0.005 × 20 = 0.1

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#### 8−3+4×5÷2 =

Follow the RULE not the acronym.... review the RULE of pemdas, u will understand why the correct answer is 15 8−3+4×5÷2 = \/ 8−3+20÷2 = \/ 8−3+10 = \/ 5 + 10 = 15

#### Four consecutive integers have a mean of 9.5. What is the largest of these integers?

Four consecutive integers could be represented as n, n+1, n+2, n+3 Therefore, by saying that they have a mean of 9.5, we mean to say: (n + n+1 + n+2 + n+3)/4 = 9.5 (4n + 6)/4 = 9.5 4n + 6 = 9.5(4) 4n = 38-6 4n = 32 n = 32/4 n = 8 Therefore, the largest value is n + 3 = 8 + 3 = 11

#### Two carpenters are building each small dog house. Carpenter A started building his at 1:00pm and ended at 5:30pm. Carpenter B, being a newbie is more slower than A by half. At what time must Carpenter B start building so that he can end at the same time with A?

Carpenter A 1:00pm-->5:30pm~>4hrs 30mins Carpenter B 1/2 of 4h30m is 2h15m So, B must be 2h15m earlier 13:00 - 2:15 = 10:45am

#### Which fraction comes next in the sequence:

1/12, 1/6, 1/4, 1/3, 5/12, __?

Solution: The above is the same as: 1/12, 2/12, 3/12, 4/12, 5/12,6/12 6/12 = 1/2

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#### The ratio of two sides of a parallelogram is 4 : 3. If its perimeter is 56 cm, find the lengths of its sides.

Solution#1: Let the lengths of two sides of the parallelogram be 4n cm and 3n cm respectively. Then, its perimeter = 2(4n + 3n) cm = 8n + 6n = 14n cm Therefore, 14n = 56 ⇔ n = ⁵⁶/₁₄ = 4. one side = (4×4) cm = 16 cm ✔ other side = (3×4) cm = 12 cm Solution#2 4 + 3 = 7 56 ÷ 7 = 8 4 × 8 = 32 ÷ 2 = 16cm ✔ 3 × 8 = 24 ÷ 2 = 12cm

#### If a = 3, then 2 / (1/7 + 1/a) = ?

2/(1/7+1/3) = 2 ÷ (3/21 + 7/21)= 2 ÷ 10/21 = 2 × 21/10 = 42/10 or 21/5

#### A boat takes 4/5 as much time to travel downstream as to its return. If the rate of the river’s current is 6 kph, what is the rate of the boat in still water?

SOLUTION: A boat takes 4/5 as much time to travel downstream as to its return. ---> 4 downstream vs 5 upstream Rate of river current that slows down the boat upstream: 6kph Let B be the speed of the boat: 4(down) = 5(up) 4(B+6) = 5(B-6) 4B + 24 = 5B - 30 4B + 24 +30 = 5B - 30 + 30 4B - 4B + 24 + 30 = 5B - 4B 24 + 30 = 5B - 4B 54 = B

#### In an election between two candidates, one got 55% of the total valid votes, 20% of the votes were invalid. If the total number of votes was 7500, the number of valid votes that the other candidate got, was

Given Total votes = 7500 one candidate = 55% of valid votes invalid votes = 20% Q: How many valid votes for the other candidate? Solution: We will solve first kng ilan yng total VALID votes. If 20% were invalid votes, meaning 80% were valid: 100% - 20% = 80% If 7500 is the total valid and invalid votes, 20% of 7500 were INvalid votes 80% of 7500 were VALID votes Since we only need the total valid votes, so we will solve it. 80% of 7500 were VALID votes 80% × 7500 = valid votes 0.8 × 7500 = valid votes 6000 = valid votes Since one candidate got 55% of the total VALID votes, so the other one got 45% 100% - 55% = 45% valid votes 45% of 6000 were valid votes of the OTHER candidate 45% × 6000 = other candidate (answer) 0.45 × 6000 = other candidate (answer) 2700 = ANSWER

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#### Which of the following is divisible (with no remainder) by 6?

divisible by 6 RULE: If the sum of the digits is divisible by 3 and the last digit is divisible by 2. So, we will focus on choices B,C,E since these were the choices with last digit divisible by 2... B.6+2+5+8+7+4=32 ✖÷3 C.1+4+7+4+1+4=21✔÷ 3 E.1+4+2+5+4+6=22 ✖÷3

#### Lyn left home and drove at the rate of 50 miles per hour for 2 hours. She stopped for lunch then drove for another 4 hours at 65 mph to reach Los Angeles. How many miles did Lyn drive to reach LA?

SOLUTION: Rate = distance/time Rate × time = distance Distance1 = 50mph × 2hr D1 = 100miles Distance2 = 65mph × 4hr D2 = 260miles D1 + D2 = answer 100m + 260m = 360miles ✔

#### In a shipment of computers, 2/25 of the computers are defective. What is the ratio of defective to non defective computers?

2/25 (two twenty-fifth) of the computer are defective ---> means on every 25 computers, 2 are defective... ---> 25-2 = 23 are not defective Ratio of DEFECTIVE vs NON 2 : 23 or 2/23

#### Which of the following numbers is divisible by 6?

Divisible by 6 are both divisible by 2 and 3. Focus on choices with even numbers and then check which one is divisible by 3

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#### 2/3 × 6 = ?

Multiplying fraction, numerator to numerator then denominator and denominator. All whole numbers are considered numerator with 1 as denominator. 2/3 × 6 = 12/3 = 4

#### If a car travels at 60 kph, what is the distance it travels in 30 minutes?

SOLUTION: Rate = distance/time Rate×time = distance Distance = 60 kph × 30min Distance = 60 kph × 0.5hr Distance = 30 km

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