Math is Fast #2
How many of these can you solve quickly (and without using pen and paper)
- A teacher gave her students 3 pieces of rambutan each and had 18 left. If she wants to give her students 5 each she would need 32 more? How many students are there in the class?
- Pedro has a total of thirty 5 and 10 pesos coins worth a total of P245. How many 10 pesos coins are there?
- A train travelling at a speed 45km faster than a tourist bus covered 33 kms at the same time the bus travelled a distance of 18km. What is the average speed of the train?
- One leg of a right triangle with a hypothenuse of length 85 mm measures 84 mm. Find the length of the other leg.
- Find a right triangle given one leg is 13.
- What is 2019 x 20202020 – 2020 x 20192019?
Here are the solutions
- A teacher gave her students 3 pieces of rambutan each and had 18 left. If she wants to give her students 5 each she would need 32 more? How many students are there in the class?
Conventional solution: Algebra
Let s = number of students
Total number of rambutan = 3s + 18 or 5s – 32
3s + 18 = 5s – 32
5s – 3s = 18 + 32
2s = 50; s = 25
MATH-Inic Alternative Solution: Reasoning
The teacher has 18 left but needs an additional 32 to give 2 more rambutan per student. So number of students is (18 +32)/2 = 25
2) Pedro has a total of thirty 5 and 10 pesos coins worth a total of P245. How many 10 pesos coins are there?
Conventional solution: Algebra
Let t = number of 10-peso coins
30 – t = number of 5 peso coins
5(30-t) + 10t = 245
150 – 5 t + 10t = 245
(10-5)t = 245 – 150
5t = 95
t = 19 – 10 peso coins
Check 19 * P10 + 11 * P5 = P190 + P55 = P245
MATH-Inic Alternative solution: Reasoning
If all 30 coins were P 5, the total would be P150. The excess of P245-150 or P95 comes from the P10 coins each contributing P5 more. So P95/5 = 19 ten peso coins.
3) A train travelling at a speed 45km faster than a tourist bus covered 33 kms at the same time the bus travelled a distance of 18km. What is the average speed of the train?
Conventional solution:
Let x = the speed of the train
x – 45 = speed of the bus
Time travelled by the bus = time travelled by the train
18/( x – 45) = 33/x
18x = 33x – 1485
-15x = – 1485
x = 99 kph
MATH-Inic Solution: Reasoning
The difference in the distance travelled by the train and the bus is 33- 18 = 15 kilometers. This is exactly 1/3 of the difference of their speed. So the speed the of train is 3 x 33 = 99kph.
4) One leg of a right triangle with a hypothenuse of length 85 mm measures 84 mm. Find the length of the other leg.
Conventional solution: long multiplication
a2 = c2 – b2 ; Pythagorean theorem
a2 = 852 – 842
a2 = 7225 – 7056
a2 = 169
a = 13mm
MATH-Inic Alternative solution: factoring
a2 = c2 – b2 ; Pythagorean theorem
a2 = (c + b) ( c – b)
a2 = (85 + 84)(85-1)
a2 = (169)(1) = 169
a = 13mm
5) Find a right triangle given one leg is 13.
Conventional solution: Trial and error: There is one equation and two unknowns, so there may be multiple solutions. Find two numbers whose squares differ by a2or a2 = c2 – b2 . We must find two numbers whose square differ by 132 or 169.
MATH-Inic Alternative Solution: Factoring
Since a2 = c2 – b2 = (c + b)( c – b)
If a2 is expressed as a product of two factors, one can be considered (c + b) and the other (c – b). Adding the two, we get ( c + b) + (c – b) = 2c or twice the hypothenuse. By solving for their difference, we get ( c + b) – ( c – b)= 2b or twice the unknown leg.
The simplest way to factor 132 or 169 is (169 x 1) so the hypotenuse is (169 + 1)/2 = 85 and the other leg is (169 – 1)/ 2 = 84.
But since (c-b) = 1, c and b are consecutive counting numbers. So the problem becomes finding two consecutive numbers adding up to 169.
6) What is 2019 x 20202020 – 2020 x 20192019?
Conventional solution: Long Multiplication/Using calculator
2019 x 20202020 – 2020 x 20192019
= 40,787,878,380– 40,787,878,380 = 0
MATH-Inic Alternative Solution: Factoring
2019 x 20202020 – 2020 x 20192019
= 2019 x (2020 x 10001) – 2020 x (2019 x10001) = 0
