Workers are digging a 99-long tunnel.

Task No.: 110443. Prototype No:
Workers are building a 42-meter-long tunnel, increasing the tunneling rate by the same number of meters every day. It is known that during the first day, workers dug 3 meters of tunnel. Determine how many meters of tunnel the workers dug on the last day if all the work was completed in 7 days.
Answer:

Task No.: 110445. Prototype No:
Workers are digging a 91-meter-long tunnel, increasing the tunneling rate by the same number of meters every day. It is known that during the first day, workers dug 10 meters of tunnel. Determine how many meters of tunnel the workers dug on the last day if all the work was completed in 7 days.
Answer:

Task No.: 110447. Prototype No:
Workers are building a 15-meter-long tunnel, increasing the tunneling rate by the same number of meters every day. It is known that during the first day, workers dug 3 meters of tunnel. Determine how many meters of tunnel the workers dug on the last day if all the work was completed in 3 days.
Answer:

Task No.: 110449. Prototype No:
Workers are building a 190-meter-long tunnel, increasing the tunneling rate by the same number of meters every day. It is known that during the first day, workers dug 10 meters of tunnel. Determine how many meters of tunnel the workers dug on the last day if all the work was completed in 10 days.
Answer:

Task No.: 110451. Prototype No:
Workers are digging a 171-meter-long tunnel, increasing the tunneling rate by the same number of meters every day. It is known that during the first day, workers dug 7 meters of tunnel. Determine how many meters of tunnel the workers dug on the last day if all the work was completed in 9 days.
Answer:

Task No.: 110453. Prototype No:
Workers are digging a 108-meter-long tunnel, increasing the tunneling rate by the same number of meters every day. It is known that during the first day, workers dug 4 meters of tunnel. Determine how many meters of tunnel the workers dug on the last day if all the work was completed in 9 days.
Answer:

Task No.: 110455. Prototype No:
Workers are digging a 143-meter-long tunnel, increasing the tunneling rate by the same number of meters every day. It is known that during the first day, workers dug 8 meters of tunnel. Determine how many meters of tunnel the workers dug on the last day if all the work was completed in 11 days.
Answer:

Task No.: 110457. Prototype No:
Workers are digging a 198-meter-long tunnel, increasing the rate of construction by the same number of meters every day. It is known that during the first day, workers dug 10 meters of tunnel. Determine how many meters of tunnel the workers dug on the last day if all the work was completed in 9 days.
Answer:

Task No.: 110459. Prototype No:
Workers are building a 55-meter-long tunnel, increasing the tunneling rate by the same number of meters every day. It is known that during the first day, workers dug 9 meters of tunnel. Determine how many meters of tunnel the workers dug on the last day if all the work was completed in 5 days.
Answer:

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Prototype of Mission B14 ( №99596 )

Two motorcyclists start simultaneously in the same direction from two diametrically opposite points on a circular track, the length of which is 14 km. How many minutes will it take for the motorcyclists to meet each other for the first time?, if the speed of one of them is 21 km/h greater than the speed of the other?

Solution

Let the speed of one motorcyclist be x km/h, then the speed of the second motorcyclist is (x+21) km/h.

Since motorcyclists start in one direction from diametrically opposite points, the initial distance between them is equal to half the length of the route, i.e. 14/2 = 7 km.

Motorcyclists are moving in the same direction, which means the speed of their approach is equal to the difference in their speeds, i.e. (x+21) - x = 21 km/h.

When the motorcyclists meet each other, one of them will travel 7 km more than the other (that is, he will travel the same amount as the other and another 7 km that was originally between them).

The time it takes one of the motorcyclists to catch up with these 7 km is

7/21 = 1/3 hour = 1/3*60 = 20 minutes.

Prototype of Mission B14 ( №99581 )

Vasya needs to solve 434 problems. Every day he solves the same number of problems more than the previous day. It is known that on the first day Vasya solved 5 problems. Determine how many problems did Vasya solve on the last day?, if he completed all the tasks in 14 days.

Solution

Let Vasya solve x more problems every day than on the previous day.

1 day: 5 tasks,

Day 2: 5+x tasks,

Day 3: 5+ x+x = 5+2x tasks,

Day 14: 5+13x tasks.

Since Vasya solved 434 problems in total, we will create and solve the equation:

5+(5+x)+(5+2x)+...+(5+13x) = 434.

On the left side of the equation is the sum of the arithmetic progression (a1 = 5, n=14, a14 = 5+13x). Then the equation will take the form:

(5+5+13x)*14/2 = 434,

(10+13x)*7 = 434,

5+13x = 57 - the number of problems that Vasya solved on the last day.

Prototype of Mission B14 ( №99580 )

Workers are building a 500-meter-long tunnel, increasing the tunneling rate by the same number of meters every day. It is known that during the first day, workers dug 3 meters of tunnel. Determine how many meters of tunnel did the workers build? on the last day if all the work was completed within 10 days.

Solution

Let the workers increase the laying norm by x meters every day.

1 day: 3 m,

Day 2: 3+x m,

Day 3: 3+ x+x = 3+2x m,

Day 10: 3+9x m.

Since the length of the tunnel is 500 m and all the work was completed in 10 days, we will compose and solve the equation:

3+(3+x)+(3+2x)+...+(3+9x) = 500.

On the left side of the equation is the sum of the arithmetic progression (a1 = 3, n=10, a10 = 3+9x). Then the equation will take the form:

(3+3+9x)*10/2 = 500,

3+9x = 97 - how many meters of tunnel the workers dug on the last day.

Prototype of Mission B14 ( №99579 )

A team of painters is painting a 240-meter-long fence, increasing the painting rate by the same number of meters every day. It is known that during the first and last days, the team painted a total of 60 meters of fence. Determine How many days did the team of painters paint the entire fence?.

Solution

Let a team of painters paint the entire fence for n days and every day increase the painting rate by x meters. If on the first day the team painted y meters of fence, then on the second day it painted - (y+x) meters of fence, and on the last, nth day - (y+(n-1)x) meters.

Since the team painted a total of 60 meters of fence on the first and last day, we get the equation:

y+(y+(n-1)x) = 60.

And since the length of the fence is 240 meters, we will create an equation:

y+(y+x)+...+(y+(n-1)x) = 240.

On the left side of this equation is the sum of n terms of the arithmetic progression (a1 = y, an = y+(n-1)x), then using the formula for the sum of the arithmetic progression the last equation will be rewritten as:

(y+y+(n-1)x)*n/2 = 240,

(y+y+(n-1)x)*n = 480

and since y+(y+(n-1)x) = 60, then substituting into the last equation, we get:

Those. a team of painters painted the fence for 8 days.

Prototype of Mission B14 ( №99578 )

There are two vessels. The first one contains 30 kg, and the second - 20 kg of solution acids of various concentrations. If these solutions are mixed, you get a solution containing 68% acid. If you mix equal masses of these solutions, you will get a solution containing 70% acid. How many kilograms of acid are contained in the first vessel?

Solution

Let vessel 1 contain x kg of acid, and vessel 2 contain y kg of acid.

1) If these solutions are mixed, you will get a solution containing 68% acid.

Since the solutions were mixed, the total mass of the solution was 20 + 30 = 50 kg. The resulting solution contains 68% acid, i.e. 68*50/100 = 34 kg of acid. We get the first equation:

2) If you mix equal masses of these solutions, you will get a solution containing 70% acid.

Let's assume that we mixed 20 kg of one solution and 20 kg of another. Then we get a solution weighing 40 kg. It contains 70% acid, i.e. 70*40/100 = 28 kg of acid.

The second solution weighing 20 kg contains y kg of acid, and the first solution contains x kg of acid per 30 kg, which means that per 20 kg there will be (x/30)*20 = (2/3)*x. Then we get the second equation:

We obtained a system of two equations:

Subtract the second from the first equation:

This means that the first vessel contains 18 kg of acid.

Prototype of Mission B14 ( №99577 )

By mixing 30% and 60% acid solutions and adding 10 kg of pure water, we obtained a 36% acid solution. If instead of 10 kg of water we added 10 kg of a 50% solution of the same acid, we would get a 41% acid solution. How many kilograms of the 30% solution were used to prepare the mixture?

two cars started in the direction. The speed of the first car is 80 km/h, and 40 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.

4. A motor ship, the speed of which in still water is 25 km/h, passes along the river and, after stopping, returns to its starting point. The current speed is 3 km/h, the stay lasts 5 hours, and the ship returns to its starting point 30 hours after departure. How many kilometers did the ship travel during the entire voyage?

5. The cyclist rode the first third of the route at a speed of 12 km/h, the second third at a speed of 16 km/h, and the last third at a speed of 24 km/h. Find the average speed of the cyclist along the entire journey. Give your answer in km/h.

6. Two dry cargo ships travel along the sea in parallel courses in one direction: the first is 120 meters long, the second is 80 meters long. First, the second cargo ship lags behind the first, and at some point in time the distance from the stern of the first cargo ship to the bow of the second cargo ship is 400 meters. 12 minutes after this, the first cargo ship lags behind the second so that the distance from the stern of the second cargo ship to the bow of the first is 600 meters. How many kilometers per hour is the speed of the first cargo ship less than the speed of the second?

7. Each of two workers with the same qualifications can complete an order in 15 hours. 3 hours after one of them began completing the order, a second worker joined him, and they completed the work on the order together. How many hours did it take to complete the entire order?

8. The first pipe passes 6 liters of water per minute less than the second pipe. How many liters of water per minute does the first pipe pass if it fills a 360-liter tank 10 minutes slower than the second pipe?

9. Five shirts are 25% cheaper than a jacket. How many percent are seven shirts more expensive than a jacket?

10. Grapes contain 91% moisture, and raisins - 7%. How many kilograms of grapes are required to produce 21 kilograms of raisins?

11. Tom Sawyer and Huckleberry Finn are painting a 100-meter long fence. Each day they paint more than the previous day, by the same number of meters. It is known that on the first and last day they painted a total of 20 meters of fence. How many days did it take to paint the entire fence?

12. Citizen Petrov had a son on August 1, 2000. On this occasion, he opened a deposit of 1000 rubles in a certain bank. Every next year on August 1, he replenished the deposit by 1000 rubles. According to the terms of the agreement, the bank accrued 20% on the deposit amount annually on July 31. After 6 years, citizen Petrov had a daughter, and he opened another deposit in another bank, already 2200 rubles, and every subsequent year he replenished this deposit by 2200 rubles, and the bank annually added 44% on the deposit amount. How many years after the birth of a son will the amounts on deposits be equal if money is not withdrawn from deposits?

day the team painted a total of 70 meters, how many days did the painters paint the entire fence?