Ratios and proportions. Percentages. Word problems
TasksTopic 2
1
Task
Matching
Two identical automatic production lines make 16 tonnes of chocolate glaze in 4 days. Match each question (1–4) with the correct answer (A–E). Assume that each line produces the same amount of glaze every day.
А
2 Б
4 В
6 Г
8 Д
12
Solution
Together the two lines produce \(16:4=4\) tonnes per day, so one line produces \(2\) tonnes per day. 1) In 3 days the two lines make \(4\cdot3=12\) tonnes — E. 2) One line needs \(16:2=8\) days to make 16 tonnes — D. 3) In 2 days one line makes \(2\cdot2=4\) tonnes — B. 4) One line makes 8 tonnes in 4 days, so \(48:8=6\) lines are needed — C. Matching: 1–E, 2–D, 3–B, 4–C.
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2
Task
Numeric answer
The positive number \(A\) is 3.8 times the positive number \(B\). By what percent is \(A\) greater than \(B\)?
Solution
Since \(A=3.8B\), we have \(A-B=2.8B\). The percentage increase is \(\dfrac{2.8B}{B}\cdot100\%=280\%\). Answer: 280.
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3
Task
One correct answer
A bank pays its depositors 8% annual interest. Determine how much money must be deposited to earn 60 UAH profit after one year.
Solution
60 UAH is 8% of the deposit \(x\): \(0.08x=60\). Hence \(x=60:0.08=750\) UAH. Answer: E.
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4
Task
One correct answer
Two specialists designed an advertisement. They were paid 5000 UAH. The first received one quarter of the money and the second received the rest. How much did the second specialist receive?
Solution
The first specialist received \(\dfrac14\), so the second received \(\dfrac34\) of the total. \(5000\cdot\dfrac34=3750\) UAH. Answer: D.
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5
Task
One correct answer
A bus and a minibus leave cities \(A\) and \(B\), which are 340 km apart by road, at the same time and travel toward each other at constant speeds of 65 km/h and 80 km/h respectively. They travel without stopping and have not yet met. Which formula gives the distance \(S\) (in km) between them after \(t\) hours?
Solution
Their closing speed is \(65+80=145\) km/h. After \(t\) hours the distance has decreased by \(145t\) km, so \(S=340-145t\). Answer: E.
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6
Task
One correct answer
During one week two couriers delivered 210 packages in total. The numbers delivered by the first and second couriers are in the ratio 3:7. How many packages did the second courier deliver?
Solution
There are \(3+7=10\) equal parts in total. The second courier has 7 parts: \(210\cdot\dfrac7{10}=147\). Answer: E.
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7
Task
Numeric answer
Petro, Mykola and Vasyl visit a café in the morning; each orders a sandwich and a hot drink. Vasyl does not drink black tea, and Mykola orders a ham sandwich. Prices: cheese sandwich 7 UAH, ham sandwich 15 UAH, fish sandwich 17 UAH; coffee with milk 13 UAH, coffee 12 UAH, black tea 8 UAH, green tea 9 UAH. Find the total cost in UAH of the cheapest possible order for all three.
Solution
For Vasyl the cheapest choice is a cheese sandwich for 7 UAH and green tea for 9 UAH, total 16 UAH. Mykola must take the 15 UAH ham sandwich and the cheapest drink, black tea for 8 UAH, total 23 UAH. Petro can take a 7 UAH cheese sandwich and 8 UAH black tea, total 15 UAH. Thus \(16+23+15=54\) UAH. Answer: 54.
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8
Task
One correct answer
A magazine cost 25 UAH. Two months later the same magazine cost 21 UAH. By what percent did the price decrease?
Solution
The price decreased by \(25-21=4\) UAH. Relative to the original price, \(\dfrac4{25}\cdot100\%=16\%\). Answer: E.
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9
Task
One correct answer
A state fee of 0.5% of the value of inherited property is charged for issuing an inheritance certificate. How much must be paid if the inherited property is worth 32,000 UAH?
Solution
0.5% of 32,000 UAH is \(32000\cdot\dfrac{0.5}{100}=160\) UAH. Answer: C.
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10
Task
Numeric answer
Parents are spending a day at an amusement park with their children Mariika (age 4) and Bohdan (age 7). Each child may visit at most three attractions, each at most once. Only children older than 6 may use the Electric Cars and Fun Hills. Bohdan will not ride the Little Train. A separate ticket is required for each child for each attraction. Ticket prices: Fun Hills 17 UAH, Little Train 16 UAH, Electric Cars 20 UAH, Carousel 12 UAH, Trampoline 15 UAH, Children's Fishing 8 UAH, Swans 13 UAH. Find the maximum total amount in UAH the parents can spend on tickets for the children.
Solution
Mariika cannot use Fun Hills or Electric Cars. Her three most expensive available attractions are Little Train 16 UAH, Trampoline 15 UAH and Swans 13 UAH: \(16+15+13=44\) UAH. Bohdan can use all attractions except the Little Train; his three most expensive are Electric Cars 20 UAH, Fun Hills 17 UAH and Trampoline 15 UAH: \(20+17+15=52\) UAH. The maximum total is \(44+52=96\) UAH. Answer: 96.
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11
Task
One correct answer
Before New Year, a household appliance store reduced the prices of all goods by 15%. How much will a television that originally cost 1800 UAH cost after the discount?
Solution
After a 15% discount, the item costs \(100\%-15\%=85\%\) of its original price. \(1800\cdot0.85=1530\) UAH. Answer: D.
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12
Task
One correct answer
If the number \(b\) is 47% of the positive number \(a\), then \(b=\)
Solution
47% of \(a\) is \(a\cdot\dfrac{47}{100}=\dfrac{a}{100}\cdot47\). In the source, options A and E are mathematically identical even though the official key gives E. The site accepts both equivalent forms.
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13
Task
Matching
A laboratory has two copper-tin alloys: the first has a mass of 50 kg and contains 10% copper; the second has a mass of 100 kg and contains 25% copper. Match each question (1–4) with the correct answer (A–E).
А
5 Б
15 В
20 Г
25 Д
30
Solution
1) The first alloy contains \(50\cdot0.1=5\) kg of copper — A. 2) The second contains \(100\cdot0.25=25\) kg, so together they contain \(5+25=30\) kg — E. 3) Total mass is 150 kg with 30 kg of copper, so \(\dfrac{30}{150}\cdot100\%=20\%\) — C. 4) Let \(x\) kg of the second alloy be added. Then \(\dfrac{5+0.25x}{50+x}=0.15\), giving \(x=25\) kg — D. Matching: 1–A, 2–E, 3–C, 4–D.
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14
Task
One correct answer
An automatically controlled machine works at a constant rate and makes 40 parts in \(t\) hours (\(t>5\)). Which expression gives the number of parts it makes in 5 hours?
Solution
In one hour the machine makes \(\dfrac{40}{t}\) parts. In 5 hours it makes \(\dfrac{40}{t}\cdot5=\dfrac{200}{t}\). Answer: E.
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15
Task
One correct answer
Flights depart from an airport every 10 minutes according to schedule. The first plane departs at 6:00 a.m. Find the scheduled departure time of the 30th plane.
Solution
There are \(30-1=29\) ten-minute intervals between the first and 30th planes: \(29\cdot10=290\) minutes = 4 h 50 min. \(6{:}00+4{:}50=10{:}50\). Answer: B.
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16
Task
Numeric answer
In one day, a citizen took out one-year loans from two banks: one at 12% annual interest and the other at 15%. The total amount borrowed was 5000 UAH. Both loans are repaid in a single payment on the final day of the agreements. The total interest charged is 654 UAH. How much money, in UAH, was borrowed at the higher interest rate?
Solution
Let \(x\) UAH be borrowed at 12% and \(y\) UAH at 15%. Then \(x+y=5000\) and \(0.12x+0.15y=654\). Substitute \(x=5000-y\): \(0.12(5000-y)+0.15y=654\). Thus \(600+0.03y=654\), so \(y=1800\). Answer: 1800.
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17
Task
Numeric answer
How many litres of a 5% salt solution must be added to 30 litres of a 12% salt solution to obtain a 9% salt solution?
Solution
Let \(x\) litres of 5% solution be added. Then \(0.05x+0.12\cdot30=0.09(30+x)\). Hence \(0.05x+3.6=2.7+0.09x\), so \(0.9=0.04x\) and \(x=22.5\). Answer: 22.5.
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18
Task
One correct answer
A 60-hectare field was planted with peas and soybeans. Peas were planted on \(\dfrac34\) of the field. How many hectares were planted with soybeans?
Milk contains 3% protein. How many grams of protein are contained in 600 g of milk?
Solution
3% of 600 g is \(600\cdot\dfrac3{100}=18\) g. Answer: B.
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20
Task
Numeric answer
It is known that \(\dfrac{y-x}{2x}=\dfrac34\), where \(0<x<y\). How many times greater is \(y\) than \(x\)?
Solution
From \(\dfrac{y-x}{2x}=\dfrac34\), we get \(4(y-x)=6x\). Thus \(4y-4x=6x\), so \(4y=10x\) and \(\dfrac yx=\dfrac{10}{4}=2.5\). Answer: 2.5.
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21
Task
One correct answer
Express the number \(\dfrac15\) as a percentage.
Solution
To express the number as a percentage, multiply by 100%: \(\dfrac15\cdot100\%=20\%\). Answer: B.
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22
Task
Numeric answer
A cyclist covers the distance between two cities in 2 hours, while a pedestrian covers it in 6 hours. Assume that both speeds remain constant throughout the journey.
1. Find the distance between the cities (in km) if the cyclist is 12 km/h faster than the pedestrian. 2. The pedestrian and cyclist start at the same time from the two cities and travel toward each other. How many hours after starting will they meet?
Solution
1) Let the pedestrian’s speed be \(x\) km/h; then the cyclist’s speed is \(x+12\) km/h. Since they cover the same distance, \(2(x+12)=6x\). Thus \(x=6\) km/h and the distance is \(6\cdot6=36\) km. 2) Their closing speed is \(6+18=24\) km/h, so the meeting time is \(36:24=1.5\) h. Answers: 1) 36; 2) 1.5.
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23
Task
Numeric answer
Determine the monthly electricity cost (in UAH) for a user entitled to a discount (see the bill fragment).
Discount (%), limit (kWh)
25% for 75 kWh
Current meter reading, kWh
6275
Previous meter reading, kWh
6160
Consumed, kWh
115
Tariff, UAH
0.28
Amount due, UAH
?
The tariff is 0.28 UAH per kWh. The benefit means that for 75 kWh of the monthly consumption, the user pays 25% less than the normal tariff cost.
Solution
For 75 kWh, the user pays 75% of the normal cost: \(75\cdot0.28\cdot0.75=15.75\) UAH. The remaining \(115-75=40\) kWh are paid at the full tariff: \(40\cdot0.28=11.20\) UAH. Total: \(15.75+11.20=26.95\) UAH. Answer: 26.95.
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24
Task
Numeric answer
The original price of a dress was 144 UAH. After a markdown, its price was reduced by 60%.
1. Calculate the price of the dress after the markdown (in UAH). 2. What percentage of the reduced price is the original price?
Solution
1) After a 60% reduction, 40% of the original price remains: \(144\cdot0.4=57.6\) UAH. 2) The original price as a percentage of the reduced price is \(\dfrac{144}{57.6}\cdot100\%=250\%\). Answers: 1) 57.6; 2) 250.
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25
Task
One correct answer
For a money transfer, a customer must pay the bank a fee equal to 2% of the transfer amount. How much money in total (in UAH) must the customer pay at the bank if the transfer amount is 30,000 UAH?
Solution
The bank fee is \(30000\cdot0.02=600\) UAH. The total payment is \(30000+600=30600\) UAH. Answer: B.
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26
Task
Numeric answer
At the Nadiia farm, 600 hectares are sown with winter wheat every year. In 2007, the average yield was 24 centners per hectare. Thanks to favorable weather conditions, in 2008 the farm harvested 19,200 centners more winter wheat than in 2007. Calculate the average yield in 2008 (centners per hectare).
(Average crop yield is the ratio of the mass harvested to the total area on which the crop was grown.)
Solution
In 2007 the harvest was \(24\cdot600=14400\) centners. In 2008 it was \(14400+19200=33600\) centners. The average yield is \(33600:600=56\) centners/ha. Answer: 56.
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27
Task
Numeric answer
A cyclist’s route is 81 km long. He covered the first part at a constant speed in 3 hours. He covered the second part, which is 36 km long, at a constant speed of 18 km/h.
1. How many hours did the cyclist spend on the second part of the route? 2. What was the cyclist’s average speed (in km/h) over the whole route?
Solution
1) Time for the second part: \(36:18=2\) h. 2) Total travel time: \(3+2=5\) h. Average speed over the whole route: \(81:5=16.2\) km/h. Answers: 1) 2; 2) 16.2.
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28
Task
One correct answer
Serhii and Petro were picking apples. Serhii picked five times as many apples as Petro. What fraction of all the apples did Petro pick?
Solution
Let Petro pick \(x\) apples; then Serhii picked \(5x\). Together they picked \(6x\), so Petro’s fraction is \(\dfrac{x}{6x}=\dfrac16\). Answer: B.
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29
Task
One correct answer
A construction company bought metal-plastic windows and doors for a new building in the ratio 4:1. Choose a number that could represent the total number of windows and doors in the building.
Solution
With a 4:1 ratio, the total consists of \(4+1=5\) equal parts, so the total must be divisible by 5. Of the given numbers, only 45 is divisible by 5. Answer: B.
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30
Task
Numeric answer
A building’s heating system requires three radiator units per 50 m³. How many radiator units must be ordered if the new building is a rectangular prism measuring 15 m × 18 m × 25 m?
Solution
The building volume is \(V=15\cdot18\cdot25=6750\) m³. Three units are needed per 50 m³, so \(6750:50\cdot3=135\cdot3=405\). Answer: 405.
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31
Task
Numeric answer
A mobile phone costs 2260 UAH in a store. The buyer cannot pay the full amount at once, so the phone is purchased in installments. Under the agreement, the buyer pays 10% of the phone price every month for 12 months.
1. Find the monthly installment (in UAH). 2. Find the total of the 12 monthly payments. By how many UAH does this total exceed the store price of 2260 UAH?
Solution
1) \(10\%\) of 2260 UAH is \(2260\cdot0.1=226\) UAH. 2) Over 12 months: \(226\cdot12=2712\) UAH. The excess is \(2712-2260=452\) UAH. Answers: 1) 226; 2) 452.
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32
Task
One correct answer
For decorating a hall, balloons of only two colors were purchased in the ratio 4:5. Which of the following numbers could be the total number of balloons purchased?
Solution
With a 4:5 ratio, the total consists of \(4+5=9\) equal parts, so it must be divisible by 9. Of the given numbers, only 117 is divisible by 9. Answer: C.
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33
Task
Numeric answer
A shoe factory produced 5340 pairs of children’s, women’s, and men’s shoes. It produced 2100 pairs of men’s shoes, and five times as many women’s shoes as children’s shoes.
1. By what percent was the number of women’s shoes greater than the number of children’s shoes? 2. How many pairs of children’s shoes were produced?
Solution
Let \(x\) be the number of pairs of children’s shoes; then women’s shoes total \(5x\). \(x+5x+2100=5340\), so \(6x=3240\) and \(x=540\). Women’s shoes exceed children’s shoes by \(4x\), which is \( rac{4x}{x}\cdot100\%=400\%\). Answers: 1) 400; 2) 540.
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34
Task
Numeric answer
A library contains only textbooks, dictionaries, reference books, and fiction. According to the pie chart, fiction makes up 75% of all books, reference books 8%, dictionaries 5%, and the remainder are textbooks.
1. Find the total number of books if there are 84 textbooks. 2. How many additional textbooks must be purchased so that the ratio of textbooks to reference books becomes 4:1?
Solution
Textbooks make up \(100-75-8-5=12\%\). If 12% equals 84 books, the total is \(84:0.12=700\). Reference books: \(700\cdot0.08=56\). For a 4:1 ratio there must be \(56\cdot4=224\) textbooks, so \(224-84=140\) more are needed. Answers: 1) 700; 2) 140.
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35
Task
Numeric answer
For 80 ninth-grade students, squared and lined notebooks are to be purchased for tests. Each student needs 9 squared notebooks and three times fewer lined notebooks. Each notebook costs 3 UAH. A 5% discount is given when notebooks are bought in packs of 10.
1. Find the total number \(N\) of notebooks needed for the 80 students. 2. How many UAH will all \(N\) notebooks cost if bought in packs of 10 with the discount applied?
Solution
Squared notebooks: \(80\cdot9=720\); lined notebooks: \(720:3=240\). Thus \(N=960\). Before the discount: \(960\cdot3=2880\) UAH. With a 5% discount: \(2880\cdot0.95=2736\) UAH. Answers: 1) 960; 2) 2736.
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36
Task
Numeric answer
Andrii deposited a certain amount of money into a payment terminal to top up his phone account. A fee of 2.40 UAH was deducted, equal to 3% of the amount deposited. The remaining amount was credited to the phone account.
1. How much money (in UAH) did Andrii deposit? 2. The mobile operator awards 8 bonus points for every 5 UAH credited to the phone account. No points are awarded for a remaining amount below 5 UAH. How many bonus points did Andrii receive?
Solution
Since 2.40 UAH is 3%, the deposited amount is \(2.40:0.03=80\) UAH. The credited amount is \(80-2.40=77.60\) UAH. This contains 15 complete groups of 5 UAH, so the bonus is \(15\cdot8=120\). Answers: 1) 80; 2) 120.
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37
Task
Numeric answer
Indian and Ceylon tea were mixed in the ratio 10:13, using 180 g of Indian tea.
1. How many grams of tea mixture were obtained? 2. By what percent is the amount of Ceylon tea greater than the amount of Indian tea?
Solution
10 parts correspond to 180 g, so one part is 18 g. Total mass: \((10+13)\cdot18=414\) g. Ceylon tea is 13 parts versus 10 parts of Indian tea, so it is greater by \( rac{13-10}{10}\cdot100\%=30\%\). Answers: 1) 414; 2) 30.
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38
Task
Numeric answer
A boat travels 24 km downstream in 5 hours and 12 km upstream in 3 hours. Find the speed of the current (in km/h). Assume that both the boat’s speed in still water and the current speed are constant.
Solution
The downstream speed is \(24:5=4.8\) km/h and the upstream speed is \(12:3=4\) km/h. From \(v+c=4.8\) and \(v-c=4\), we get \(2c=0.8\), so \(c=0.4\) km/h. Answer: 0.4.
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39
Task
Numeric answer
An exhibition contains only two types of artworks: paintings and sculptures. The number of sculptures is four times smaller than the number of paintings.
1. What percentage of all artworks are paintings? 2. By what percent is the number of paintings greater than the number of sculptures?
Solution
Let the number of sculptures be \(x\); then there are \(4x\) paintings and \(5x\) artworks in total. Paintings make up \( rac{4x}{5x}\cdot100\%=80\%\). Paintings exceed sculptures by \(3x\), which is \( rac{3x}{x}\cdot100\%=300\%\). Answers: 1) 80; 2) 300.
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40
Task
Numeric answer
There are two salt-water solutions. The salt concentration of the first is 0.25 and of the second is 0.4. By how many kilograms must the amount of one solution exceed the amount of the other in order to obtain 50 kg of a solution with salt concentration 0.34?
Solution
Let \(x\) kg of the first solution be used; then \(50-x\) kg of the second are used. \(0.25x+0.4(50-x)=0.34\cdot50\). This gives \(x=20\), so 30 kg of the second solution are used. The difference is \(30-20=10\) kg. Answer: 10.
