Time : 50 minutes
Answer : End of the problem
Problem 1
Problem 2
(2)
(3)
(4)
(5)
(6)
Problem 3
Problem 4
(1) Calculation
2/3 - (5/6 - 4/9) / 7/3 =
(2) Calculation
1.12 × 3.2 + 1.12 × 2.7 - 1.12 × 0.9 =
(3) Find X
15 - {37 - 4 × (9 - X)} = 6
(4) Find A, B and C
47/18 hours = A hours , B minutes and C seconds
Problem 2
(1)
The ratio of 3 integers is 2 : 3 : 4 and the least common multiple is 144.
Find the sum of these 3 integers.
(2)
I bought same number of sheets of 50 yen stamp and 80 yen stamp, respectively.
The difference of the each total price was 540 yen.
How many sheets did I buy all together?
(3)
It's 770 yen for 3 apples and 2 peaches.
It's 1950 yen for 5 apples and 6 peaches.
How much is it for 1 apple and 1 peach?
(4)
Hanako is solving problems of Sansue workbook every day.
She finished solving 72% of the number of all problems today.
If she solves 4 more problems, she would finish 80 %.
How many problems are there in all?
(5)
When 4 bottles of water are distributed to each member of a certain soccer team, 8 bottles remain.
When 5 bottles are distributed, 8 bottles run short.
Find the number of people of the team and the number of the bottle of water.
(6)
Find the area of parallelogram ABCD of the figure below.
Problem 3
The following sequence is lined in accordance with certain rules.
(1) Find the fourth number from the left of the seventh step.
(2) Find the second number from the right of the twelfth step.
(3) Find the sum total of numbers of the twelfth step.
Problem 4
Between the mountain top and foot of 3600 m two cable cars of A and B is scheduled to be operated as graph below.
It takes 32 minutes for descent and 40 minutes for ascent for both cable cars and they advance with a fixed speed.
Answer the following questions.
(1) As scheduled, A and B left at the same time.
How many meters from the foot do 2 cable cars pass each other first?
(2) Since it took time for checking, B was late to start descending for 10 minutes.
How many meters from the foot do 2 cable cars pass each other first in this case?
(3) The place where 2 cable cars pass each other was the spot in the middle of the top and foot of the mountain.
Answer which cable car was late to start in this case.
How many minutes was it late for leaving?
<Answer>
Problem 1
Problem 2
(2)
(3)
(4)
(5)
(6)
Problem 3
Problem 4
(1) Calculation
2/3 - (5/6 - 4/9) / 7/3 =
(2) Calculation
1.12 × 3.2 + 1.12 × 2.7 - 1.12 × 0.9 =
(3) Find X
15 - {37 - 4 × (9 - X)} = 6
(4) Find A, B and C
47/18 hours = A hours , B minutes and C seconds
Answer
(1) 1/2
(2) 5.6
(3) X = 2
(4) A = 2, B = 36, C = 40
Problem 2
(1)
The ratio of 3 integers is 2 : 3 : 4 and the least common multiple is 144.
Find the sum of these 3 integers.
Answer
108
Solution
3 integers are expressed as 2 × A, 3 × A, 4 × A.
The least common multiple calculated as below is A × 2 × 1 × 3 × 2 = 144.
Thus A = 144 / 12 = 12.
Therefore the sum of 3 integers = 24 + 36 + 48 = 108.
(2)
I bought same number of sheets of 50 yen stamp and 80 yen stamp, respectively.
The difference of the each total price was 540 yen.
How many sheets did I buy all together?
Answer
36 sheets
Solution
The difference of one sheet = 80 - 50 = 30 yen.
The number of sheet = 540 / 30 = 18.
Therefore total number = 18 × 2 = 36 sheets.
(3)
It's 770 yen for 3 apples and 2 peaches.
It's 1950 yen for 5 apples and 6 peaches.
How much is it for 1 apple and 1 peach?
Answer
340 yen
Solution
It's 2310 yen for 9 apples and 6 peaches.
Then the difference of the price of 9 apples + 6 peaches and 5 apples + 6 peaches = 2310 - 1950 = 360 yen which is the price of 9 - 5 = 4 apples.
Thus 1 apple = 360 / 4 = 90 yen.
1 peach = (770 - 90 × 3) / 2 = 250 yen.
Therefore 1 apple + 1 peach = 90 + 250 = 340 yen.
(4)
Hanako is solving problems of Sansue workbook every day.
She finished solving 72% of the number of all problems today.
If she solves 4 more problems, she would finish 80 %.
How many problems are there in all?
Answer
50 problems
Solution
4 problems are equivalent to 80 - 72 = 8 % of all problems of the workbook.
Therefore the number of problem is 4 / 0.08 = 50 problems.
(5)
When 4 bottles of water are distributed to each member of a certain soccer team, 8 bottles remain.
When 5 bottles are distributed, 8 bottles run short.
Find the number of people of the team and the number of the bottle of water.
Answer
16 persons
Solution
It'll be 8 + 8 = 16 bottles need to increase 1 bottle of the water distributed to each person.
Therefore the number of people is 16/1 = 16 persons.
The number of the bottle of water is 4 × 16 + 8 = 72.
(6)
Find the area of parallelogram ABCD of the figure below.
Answer
32 cm2
Solution
∠BAC= 180 - 75 × 2 = 30 degrees As shown in the figure below, AC : CE = 2 : 1.
Thus CE = 8/2 = 4 cm.
Therefore the area of parallelogram ABCD = 8 × 4 / 2 × 2 = 32 cm2.
Problem 3
The following sequence is lined in accordance with certain rules.
(1) Find the fourth number from the left of the seventh step.
(2) Find the second number from the right of the twelfth step.
(3) Find the sum total of numbers of the twelfth step.
Answer
(1) 20
(2) 11
(3) 2048
Solution
(1) As shown in the figure below, 4th number from the left of the 7th step is 20.
(2) The 2nd number from the right is to be the number of the step minus one.
Therefore it i s 12 - 1 =11.
(3) The sum total of numbers of each step can be calculated as multiply 2 times of the number of step minus one.
Therefore the sum total of numbers of 12th step is 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2048.
Problem 4
Between the mountain top and foot of 3600 m two cable cars of A and B is scheduled to be operated as graph below.
It takes 32 minutes for descent and 40 minutes for ascent for both cable cars and they advance with a fixed speed.
Answer the following questions.
(1) As scheduled, A and B left at the same time.
How many meters from the foot do 2 cable cars pass each other first?
(2) Since it took time for checking, B was late to start descending for 10 minutes.
How many meters from the foot do 2 cable cars pass each other first in this case?
(3) The place where 2 cable cars pass each other was the spot in the middle of the top and foot of the mountain.
Answer which cable car was late to start in this case.
How many minutes was it late for leaving?
Answer
(1) 1600 m
(2) 2100 m
(3) B delayed 4 minutes
Solution
(1) The time ratio of ascent and descent is 40 : 32 = 5 : 4.
Because the speed ratio is to be inverse ratio of the time ratio and ascent : descent = 4 : 5.
Since the moving distance is proportional to the speed, the distance from the foot where 2 cable cars passed each other is 3600 × 4/(4+5 ) = 1600 m.
(2) The upper triangle and the lower triangle are homothetic in a figure below and the homothetic ratio is (40- 10) : 42 = 5 : 7.
Accordingly the distance from the foot is 3600 × 7/(5 + 7) =2100 m.
(3) According to the result of (1) and (2), when departure of B is behind schedule, it is found out that the distance from the foot becomes longer than the case of leaving as scheduled.
Therefore it's B that was behind the schedule in case the place passing each other is the middle of the top and foot.
It is assumed that B was late to start ① minutes.
In the figure below, the upper triangle and the lower triangle become congruent.
40 - ① = 32 + ①, then ② = 8.
Therefore ① = 4 and B was late for 4 minutes.
