Question

1.(12/7-20/21)*56/16 2.Девочка прочитала книгу за 360 минут. Сколько времени она читала 2/3 книги 3.Вычислите:3.29+0.6*1.3 4.7/15 разделить (6/10-8/3)+19*2/31 5.Фома, выполняя свою домашнюю работу, тратит на выполнение задания по математике 60% времени, 20% от оставшегося времени у него уходит на задание по литературе. Определите, сколько всего времени уходит у Фомы на выполнение домашнего задания, если на выполнение задания по литературе у него уходит 40 минут. Ответ укажите в минутах. 6.Вася и Петя выписывают 12-значное число, ставя цифры по очереди, начиная со старшего разряда. Докажите, что какие 6ы цифры он не писал, Петя всегда сможет добиться, чтобы получившееся число делилось на 4.

Answer 1

1.(12/7-20/21)*56/16 = 2 2/3
             1) 12/7 - 20/21 = 36/21 - 20/21 = 16/21
             2) 16/21 * 56/16 = 8/3 = 2 2/3
2.Девочка прочитала книгу за 360 минут. Сколько времени она читала 2/3 книги:  360 * 2/3 = 360 : 3 * 2 = 240 минут = 4 часа
3.Вычислите:3.29+0.6*1.3 = 3,29 + 0,78 = 4,07
4.7/15 разделить (6/10-8/3)+19*2/31 = 1
          1) 6/10 - 8/3 = 18/30 - 80/30 = - 62/30 = - 31/15
          2) 7/15 : (- 31/15) = 7/15 * (- 15/31) = - 7/31
          3) 19 * 2/31 = 38/31
          4) - 7/31 + 38/31 = 38/31 - 7/31 = 31/31 = 1
5.Фома, выполняя свою домашнюю работу, тратит на выполнение задания по математике 60% времени, 20% от оставшегося времени у него уходит на задание по литературе. Определите, сколько всего времени уходит у Фомы на выполнение домашнего задания, если на выполнение задания по литературе у него уходит 40 минут. Ответ укажите в минутах.
      1) 40 мин : 20% = 40 : 0,2 = 200 минут тратит на все д/з
     

Answer 2

Let's solve each of the given problems step by step:

1. (12/7 - 20/21) * 56/16 To simplify this expression, first perform the operations inside the parentheses: (12/7 - 20/21) = (123/73 - 201/211) = (36/21 - 20/21) = 16/21 Now, multiply the result by 56/16: (16/21) * (56/16) = (1656)/(2116) = 56/21 = 2.6667 (approximately)

2. The girl read the book in 360 minutes. She read 2/3 of the book. To find out how much time she spent reading 2/3 of the book, multiply the total time by 2/3: 360 * 2/3 = 240 minutes

3. Calculate: 3.29 + 0.6 * 1.3 Simply perform the multiplication first: 0.6 * 1.3 = 0.78 Then add it to 3.29: 3.29 + 0.78 = 4.07

4. Calculate: 7/15 ÷ (6/10 - 8/3) + 19 * 2/31 To simplify the expression inside the parentheses, first find the common denominator for (6/10 - 8/3), which is 30: 6/10 = 18/30 8/3 = 80/30 Now, (6/10 - 8/3) = (18/30 - 80/30) = (-62/30) = -31/15

Now, perform the division: 7/15 ÷ (-31/15) = (7/15) * (-15/31) = -7/31

Finally, calculate: -7/31 + 19 * 2/31 = (-7 + 38)/31 = 31/31 = 1

5. Foma spends 60% of his time on math homework, and 20% of the remaining time on literature homework. He spends 40 minutes on literature homework. We need to find the total time he spends on homework.

Let T be the total time. Foma spends 60% of T on math, leaving 40% of T for other subjects (including literature). He spends 20% of this 40% on literature.

Math time: 0.60 * T Literature time: 0.20 * 0.40 * T = 0.08 * T

Total time on homework: Math time + Literature time Total time = 0.60 * T + 0.08 * T = 0.68 * T

We know that the total time is 40 minutes: 0.68 * T = 40 T = 40 / 0.68 ≈ 58.8235

So, the total time Foma spends on homework is approximately 58.82 minutes.

6. Let's prove that no matter which 6 digits Vasya writes, Petya can always add one more digit to make the 12-digit number divisible by 4.

Any number is divisible by 4 if the last two digits of the number are divisible by 4. To prove that Petya can always choose one digit to make the 12-digit number divisible by 4, we need to consider the last two digits:

Case 1: If the last two digits are not divisible by 4: In this case, there are exactly 25 two-digit numbers that are divisible by 4 (00, 04, 08, ..., 96). No matter which 6 digits Vasya writes in front of these last two digits, Petya can always choose one digit to make the 12-digit number divisible by 4.

Case 2: If the last two digits are divisible by 4: In this case, there are 25 two-digit numbers that are divisible by 4 (00, 04, 08, ..., 96), plus 3 more (12, 16, 32) that are divisible by 4 and consist of the digits 1, 2, and 3. So, there are 28 two-digit numbers in total that are divisible by 4.

If Vasya chooses any 6 digits, it is always possible to find one digit from the set {0, 1, 2, 3} such that the last two digits of the 12-digit number are divisible by 4. This is because no matter what the 6 digits are, there will be at least one two-digit number in the form XY (where X and Y are two different digits) among the chosen digits. Since there are three two-digit numbers in the set {12, 16, 32}, Petya can choose one of these numbers to make the 12-digit number divisible by 4.

Therefore, Petya can always add one more digit to the 6 digits chosen by Vasya to make the resulting 12-digit number divisible by 4.