Forming a linear equation

Q. Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award ₹x each, ₹y each and ₹z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹2, 200. School Q wants to spend ₹3, 100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each values is ₹1, 200, using matrices, find the award money for each value.
Apart from these three values, suggest one more value which should be considered for award.

Q. Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs. x, y and z respectively per person. The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then using matrix method find the value of x, y and z. What values are described in this equations?

Q. A pizza delivery worker purchased a used motor scooter that had been driven 12100 miles. He drives the motor scooter only on days he is working, during which he drives an average of 50 miles per day. After d days of pizza delivery, the motor scooter has been driven a total of 25000 miles. Which of the following equations best models this situation?

1. (12100 + d) 50 = 25000
2. 50 (12100 + d) = 25000
3. 12100d + 50 = 25000
4. 12100 + 50d = 25000

Q. Phytoremediation is the use of plant growth to purify pollutants from soil, water, or air. Suppose that a crop of brake ferns can remove 15 milligrams per square meter of a particular pollutant from the soil in 20 weeks. After, 20 weeks the ferns are harvested and a new crop is planted. If C represents the number of corps of brake ferns needed to phytoremediate soil contaminated with 170 mg per square meter of the pollutant down to healthy levels of 5 mg per square meter, which equation best models the situation ?

1. $170-20C=5$
2. $170+20C=5$
3. $170-15C=5$
4. $170+15C=5$

Q. Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹x each, ₹y each and ₹z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹1, 000. School Q wants to spend ₹1, 500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

Q.

$228$ logs are to be stacked in a store in the following manner: $30$ logs in the bottom, $28$ logs in the next row, then $26$ and so on, in how many rows can these $228$ logs be stacked? How many logs are there in the last row?

Q. Jack and Jill are filling up a small pool with water. For every two pails that Jack fills, Jill fills exactly one. If both of them fill the pool, with total of 36 pails, how many pails did Jack and Jill each contribute?

1. (24 ; 12)
2. (20 ; 16)
3. (18 ; 18)
4. (12 ; 24)

Q. Dimitri is helping to plan the school talent show. Each performer for the talent show has 6 minutes for his or her performance, which includes translation time between performances. If the introduction for the talent show is 24 minutes long and the entire show, including the introduction, will last 150 minutes, how many different performances can the talent show accommodate?

1. 21
2. 24
3. 25
4. 29

Q.

Dalia is installing a tile floor in a room. Dalia has 152 tiles available to tile the room. If each row requires $9\frac{1}{2}$ tiles and 19 tiles break while Dalia is laying the floor, how many rows of unbroken tile can she install before running out of tiles?

1. 12
2. 16
3. 14
4. 18

Q. A lift company installs service lifts in new buildings. The number of meters of cable needed for a lift is 12 times the number of floors in the building. Plus an extra 16 meters needed to go into the machinery. Which of the following function best describes how many meters of cable are needed to install the lift if there are F floors in the building ?

1. y = 16F + 12
2. y = 12F + 12
3. y = 16F + 16
4. y = 12F + 16

Q.

Find two positive numbers x and y such that their sum is 35 and the product x²y⁵ is a maximum

Q. Bolt is preparing himself to participate in a marathon by running daily 21 miles. One fine day, after running 10 miles, he takes a break and resumes running at the rate of 9 miles per hour for $n$ minutes. Which of the following equation best models the situation?

1. $10+9n=21$
2. $10=21+9n$
3. $10+9\left(\frac{n}{60}\right)=21$
4. $10=21+9\left(\frac{n}{60}\right)$

Q. There are two types of fertilizers F₁ and F₂. F₁ consists of 10% nitrogen and 6% phosphoric acid and ​F₂ consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds the she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F₁ costs ₹6/kg and F₂ costs ₹5/kg, determine how much of each type of fertilizer should be used so that the nutrient requirements are met at minimum cost. What is the minimum cost?

Q. John has taken up a project for fixing wallpaper on the walls of all the room in a hotel. Each wall requires $7\frac{1}{4}$ of paper to cover the entire wall. John had 628 wallpaper in total and got 19 wallpaper torn in the entire process. Assuming that each room in the hotel are identical and have exactly four walls to cover, then how many rooms does the hotel have?

1. 21

2. 42
3. 84
4. 29

Q.

A firm produces x type A and y type B bags. Demand for bag B is atmost one fourth of A. The corresponding constraint is

1. x < 4y

2. 

3. 4y – x ≥ 0

4. x – 4y ≥ 0

Q. Twelve years hence a man will be four times as he was 12 years ago, then his present age is

1. 25 years
2. 20 years
3. 28 years
4. 30 years

Q. $120$ kites were divided equally among a certain number of children in a school. If four children were less, each would have got $1$ kite more. Find the number of children.

Q. $\frac{25\%of50\%of100\%}{25of100\times 50\%of100}$ is equal to ______ .

1. $0.01\%$
2. $1\%$
3. $0.1\%$
4. $0.0001\%$

Q. Nitrogen fixation is the process where some plants like leguminous plants will fix nitrogen level in soil by converting the nitrogen from atmosphere into nitrate in soil. Suppose that a crop of pulses can add 25 pounds of nitrogen per square hectare in 12 weeks. After, 12 weeks the pulses are harvested and a new crop is planted. If k represents the number crop of pulses required to increase the nitrogen on a land of malnutrition soil with 160 pounds per hectare square to a healthy 335 pounds per hectare, which of the following equation best models the situation?

1. 160 - 25k = 335
2. 160 + 25k = 335
3. 160 - 12k = 335
4. 160 + 12k = 335

Q. A tin of oil was 4/5 full.When 6 bottles of oil were taken out and 4 bottles of oil were poured into it, it was 3/4 full.How many bottles of oil can the tin contain?

Q. A man sells home-made pizzas at $\$8$ per pizza. He invested an initial$\$30$ dollars for making pizzas. Write an expression for his profit if he sells N pizzas.

1. 30 - 8N
2. 30 + 8N
3. 30N - 8
4. 8N - 30

Q. A factory makes two types of items $A$ and $B$, made of plywood. One piece of item A requires $5$ minutes for cutting and $10$ minutes for assembling. One piece of item $B$ requires $8$ minutes for cutting and $8$ minutes for assembling. There are $3$ hours and $20$ minutes available for cutting and $4$ hours for assembling. The profit on one piece of item $A$ is Rs $5$ and that on item $B$ is Rs $6$. How many pieces of each type should the factory make so as to maximise profit? Make it as an L.P.P. and solve it graphically

Q. Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs. x, y and z respectively per person . The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then using matrix method find the value of x, y and z.

Q. A plumber charges a one time visit fee and then charges on hourly basis plus the rate for any part/material needed plus the tax on the required material value. If the total cost of completing a job that takes $h$ hours is given by the function $C\left(h\right)=45h+1.06\left(82.5\right)+75$, then the term $1.06\left(82.5\right)$ represents

1. the site visit fee
2. the hourly rate
3. the cost of the materials, not including tax
4. the cost of the materials, including tax

Q. There are two types of fertilisers F₁ and F₂. F₁ consists of 10% nitrogen and 6% phosphoric acid and F₂ consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F₁ costs Rs 6/kg and F₂ costs Rs 5/kg, determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

Q. Sam and his friend Toms combines weight is 150 kg. Sams weight is 84 kg less than twice of Toms weight. How much does Tom weigh? Solution with algebraic equation.