Fraction apportioning

[Ams627]

A financial sum in crowns was prepared for the rewards for the three best competitors. The first contestant won half of this amount. The second contestant received 300 crowns. The third contestant won the rest of the prepared amount, which was three times less crowns than the first one won contestant

how many times more crowns did the second contestant get than the third contestant,

how many crowns were prepared for rewards in total.

[Ams627]

A competitor ran the entire route in 3 hours. He covered a third of the entire route in the first hour. During the last hour, he had covered only 9 km, which was a quarter of the entire route.

Calculate how many km the competitor ran during the second hour.

Solution: First hour a third = 4/12, second hour a quarter = 3 / 12. So there are 5/12 remaining which is the proportion of the route he ran in the second hour. The entire route is 36km. 5/12 is 15km.

Answer: 15km.

[Ams627]

Varation: A runner in a competition ran the route in 4 hours. In the last hour he ran 8km which is one sixth of the entire route. In the first hour he ran a third of the route, then in the second hour a quarter of the route. How many km did he run in the third hour?

Solution: Entire route is 48km. He ran 1/3 + 1/4 + 1/6 which is (4 + 3 + 2) / 12 or 9/12. So he ran 1/4 in the third hour too. Check: 16 + 12 + 12 + 8 = 48km.

Answer: 12 km.

[Ams627]

Variation: A runner in a competition ran the route in two and a half hours. In the final half hour he ran 6km which is one sixth of the route. In the first hour he ran half of the route. How many km did he run in the second hour?

Entire route is 36 km. 1/4 of the route is 7.5 km. Known parts are 1/2 + 1/6. So 4/12 remaining. 4/12 = 12.

Answer: 21km.

[Ams627]

One large and one medium lie on the scales and three equal small weights. The weight of the medium weight is one third less than the weight of the large weight. One large and one small weight weigh 100g together, as do one medium and two small weights.

Determine:

1. How many times greater is the mass of the large weight than the mass of the small weight?
2. How many grams does the middle weight weigh?

Solution: We can remove one small weight from each side. So L = M + S. We know that the medium weight is two thirds of the large weight, so the small weight must be one third. So on the left of the scales we have a number plus a third of the number making 100g. The number is 75. The medium weight is thus 50g and the small weight is 25g.

[Ams627]

There are several rooms in the holiday cottage. In one room there are 2 beds and in each of the other rooms there are 3/10 of all the beds that are in the holiday cottage.

Determine: 5.1 the number of all beds in the holiday cottage, 5.2 the number of rooms in the holiday cottage.

[Ams627]

After the spring break, many students at the school gradually fell ill.

On Monday one sixth of all the students where missing. On Tuesday a quarter of all the students were missing. On Friday at school, there were only 80 students which is one third of all the pupils in the school.

1. How many students does the school have?
2. How many students were in the school on Monday?
3. How many more sick pupils were there on Friday than on Tuesday??

Solution:

1. A third of the students were there and Friday and there were 80 of them. So there are 240 students in the school.
2. There were five sixths of the students present on Monday. One sixth is 40, so there were 200 students present.
3. A quarter were sick on Tuesday which is 60 students. 160 studentsd were sick on Friday which is 100 more.

[Ams627]

Matěj walked the entire scenic route that leads from the station to the lake.

• He walked from the station to the first viewpoint 1/6 of the route.
• After another 5.5 km of walking, he reached the second view point.
• From the second viewpoint to the lake is 2/9 of the entire route.
• 1 km before the second viewpoint, Matěj stopped at a well.

Calculate how many km Matěj walked from the station to the first viewpoint. 6.2 Express as a fraction how much of the route Matěj walked from the station to the well. Write the fraction in basic form.

In the record sheet, indicate the solution procedure in both parts of the problem.

Solution: The whole route is 1/6 + 5.5km + 2/9, or 3/18 + 5.5km + 4/18. So 5.5km is 11/18, and 1/18 is half a km. So the whole route is 9km and the distance from the station to the first view point is 1.5km.

From the station to the second view point is 1.5 + 5.5, so 7km. To the well is 6km, which is 2/3 of the route.

[Ams627]

Pavel, Rosťa and Sofie signed up as a three-member relay team for a 36 km long charity run. They divided the running route into three sections of different lengths. However, Rosťa fell ill, so Pavel ran half of his section and Sofia ran the other half. In fact, Pavel ran a third longer distance than he was originally supposed to run, and Sofia a quarter longer distance than she was originally supposed to run. Calculate how many km Rosťa was originally supposed to run.