Forming simultaneous linear equations

Q.

A couple went for a picnic, they have five sons and each son has three sisters. Each sister has one baby. In total, how many people went for the picnic?​

Q. A total amount of ₹7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and $8\frac{1}{2}$% respectively. The total annual interest from these three accounts is ₹550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices.

Q.

A manufacturer produces three products x, y, z which he sells in two markets.

Annual sales are indicated below:

Market	Products
I	10000	2000	18000
II	6000	20000	8000

(a) If unit sale prices of x, y and z are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.

Q. In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paisa) is given in matrix A as

$A=\left[\begin{array}{c}140\\ 200\\ 150\end{array}\right]\begin{array}{c}\mathrm{Telephone}\\ \mathrm{House}\mathrm{calls}\\ \mathrm{Letters}\end{array}$

The number of contacts of each type made in two cities X and Y is given in the matrix B as

$$\begin{gathered} \begin{array}{ccc}\mathrm{Telephone}& \mathrm{House}\mathrm{calls}& \mathrm{Letters}\end{array} \\ B=\left[\begin{array}{ccc}1000& 500& 5000\\ 3000& 1000& 10000\end{array}\right]\begin{array}{c}\mathrm{City}X\\ \mathrm{City}Y\end{array} \end{gathered}$$

Find the total amount spent by the party in the two cities.

What should one consider before casting his/her vote − party's promotional activity of their social activities?

Q. A boat traveled 210 miles downstream and back. The trip downstream took 10 hours. The trip back took 70 hours. What is the speed of the boat in still water?

1. 9 mph
2. 13 mph
3. 8 mph
4. 12 mph

Q. Two factories decided to award their employees for three values of (a) adaptable tonew techniques, (b) careful and alert in difficult situations and (c) keeping clam in tense situations, at the rate of ₹ x, ₹ y and ₹ z per person respectively. The first factory decided to honour respectively 2, 4 and 3 employees with a total prize money of ₹ 29000. The second factory decided to honour respectively 5, 2 and 3 employees with the prize money of ₹ 30500. If the three prizes per person together cost ₹ 9500, then
i) represent the above situation by matrix equation and form linear equation using matrix multiplication.
ii) Solve these equation by matrix method.
iii) Which values are reflected in the questions?

Q.

You earn $\$9.20$ per hour at your summer job. Write and solve an inequality that represents the numbers of hours you can work to earn enough money to buy a smart phone that costs $\$299$.

Q.

A picnic is being planned in a school for Class$VII.$ Girls are $60\%$ of the total number of students and are $18$ in number. The picnic site is $55km$from the school and the transport company is charging at the rate of $Rs12perkm$. The total cost of refreshments will be $Rs4280.$ Find the cost per head if two teachers are also going with the class?

Q.

In one basketball game, a player scored $31$ points with a combination of two-point baskets, three-point baskets, and one-point free throws.

She made $5$ more two-point baskets than one-point free throws and $3$ times as many two-point baskets as three-point baskets.

How many three-point baskets, two-point baskets, and free throws did the player make?

Q.

Amanda works as a carpenter and as a blacksmith.
She earns $\$20$ per hour as a carpenter and $\$25$ dollar per hour as a blacksmith. Last week, Giselle worked both jobs for a total of 30 hours, and earned a total of $\$690$dollar. How long did Giselle work as a carpenter last week, and how long did she work as a blacksmith?

1. Carpenter : 15 hrs ; Blacksmith : 15hrs
2. Carpenter : 10 hrs ; Blacksmith : 20 hrs
3. Carpenter : 19 hrs ; Blacksmith : 11hrs
4. Carpenter : 12 hrs ; Blacksmith : 18hrs

Q. James can bake 2 muffins in 7 minutes. Alex can bake 4 muffins in 15 minutes. James starts baking muffins at 1.30 p.m and Kylie joins him at 1.45 p.m. If both of them work straight through at the above rates, at what time will they finish baking the 54th muffin ?

1. 3 : 00 p.m
2. 3 : 12 p.m
3. 3 : 15 p.m
4. 3 : 30 p.m

Q. John has a total of $\$4.30$ in dimes (1 dime $=\$0.10$) and quarter (1 quarter $=\$0.25$), and he has 19 coins in total. Which of the following systems of equations can be used to find the number of dimes, $d$ and the number of quarters, $q$ he has?

1. $d+q=19$
   $0.1d+0.25q=4.30$
2. $d+q=4.30$
   $0.1d+0.25q=19$
3. $0.1d+0.25q=19$
   $10d+25q=4.30$
4. $0.1d+0.25q=19$
   $25d+10q=4.30$

Q. A store manager calculates his store's monthly utility expenses using two expense rates $r_1$ for the dollar cost per hour the store was open during the month and $r_2$ for the dollar cost per hour the store was not open during the month. During November, which has 30 days, the store was open 8 hours a day, except one day when it was open for 15 hours for a special sale. Which of the following expressions should the manager use to calculate the store's utility expenses, in dollars for November ?

1. $\left(247r_1\right)\left(473r_2\right)$
2. $473r_1+247r_2$
3. $\frac{473r_1}{247r_2}$
4. $247r_1+473r_2$

Q. Albert has two types of shirts in his wardrobe - long sleeved and short sleeved. The number of long sleeved shirts ' l ' he has is one more than twice the number of short sleeved shirts, ' s '. If he has a total of 40 shirts in his wardrobe, how many short sleeved shirts does he have?

1. 11
2. 13
3. 17
4. 21

Q. There are some lotus flowers in a pond and some bees are hovering around. If one bee lands on each flower, one bee will be left. If two bees land on each flower, one flower will be left. Then, the number of flowers and bees respectively are ______.

1. 4, 3
2. 3, 4
3. 2, 3
4. 3, 2

Q. ''Great bakers'' want to bake a cake to make a world record. They decided to make a rectangular cake where the length of the cake should be three times it's breadth and with a perimeter 120 feet. How long will be the cake?

1. 30
2. 25
3. 45
4. 60

Q.

There are $88$ numbers $a_1,a_2,a_3,\dots ,a_{88}$ and each of them is either equal to $-3$ or $-1.$ Given that $a_1^2+a_2^2+\cdots +a_{88}^2=280,$ then the value of $\frac{a_1^4+a_2^4+\cdots +a_{88}^4}{4}-500$ is
(correct answer + 3, wrong answer 0)

Q. Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A 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 ₹1, 600. School B wants to spend ₹2, 300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹900, 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. Ross has two types of boxes. The number of mini boxes, $m$ is 1 fewer than twice the number of big boxes, $l$. If he has 46 boxes in total, which of the following systems of equations can be used to correctly solve for $m$ and $l$?

1. $m=2l-1$
   $m+l=46$
2. $m=2l-1$
   $m=l+46$
   
3. $l=2m-1$
   $m+l=46$
   
4. $l=2m-1$
   $m=l+46$

Q. A charity is planning a concert to raise money. There are 135 stall tickets and 70 deluxe tickets. The cost of deluxe tickets is 20 percent more than a stall ticket plus an additional $1.50\$$. The concert organizing committee expects to sell all the tickets and raise $2750\$$ from the ticket sales. Which of the following system of equations can be used to determine the price, s of each stall tickets and the price, d, of each deluxe ticket ?

1. 70d + 135s = 2750 ; d - 1.50s = 1.20
2. 70d + 135s = 2750 ; d - 1.20s= 1.50
3. 135s + 70d = 2750 ; d - 1.50s = 1.20
4. 135s + 70d = 2750 ; d - 1.20s = 1.50

Q. A manufacturer of TV sets produced $600$ units in the third year and $700$ units in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find the production in

Q. Joey is planning to join a local library and is researching his options. His first option, British Library, charges a one-time enrollment fee of $\$60.00$ plus an additional monthly fee of $\$18.00$. His second option, Just Book Library, does not have an enrollment fee, but charges $\$30.00$ per month. In how many months will the total cost of a library membership at both libraries be equal?

1. 6
2. 3
3. 5
4. 15

Q. The school book store sold 8 more pencils than pens one day. The cost of a pencil is $\$.05$, and the cost of a pen is $\$.20$. If the day's sales of pens and pencils totalled $\$8.90$, how many pencils were sold?

1. 39
2. 42
3. 35
4. 45

Q.

Adam has $ 1.50. He only has quarters and nickels with a total of 14 coins. How many quarters does he have? [ 1 quarter = 1/4*1 dollar; 1 nickel = 1/20* 1 dollar ]

1. 5
2. 2
3. 7
4. 4

Q. In a four digit number, the sum of the first two digits is equal to that of the last two digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other two digits. What is the third digit of the number?

1. $5$
2. $8$
3. $1$
4. $4$

Q. Robin breaks his piggy bank and counts to see a total of 23 coins adding to $2.75\$$. Of the total coins, m coins are worth $\$0.25$ each and n coins are worth $\$0.05$ each. If he did not put any other type of coin in the piggy bank, then which of the following pairs of equation, best describes the given situation?

1. m + n = 23 ; mn = 2.75
2. m + n = 23; 0.25n + 0.05m = 2.75
3. m + n = 23; 0.25m + 0.05n = 2.75
4. 0.25m + 0.05n = 23 ; m + n = 2.75

Q. The perimeter of a rectangular field is 138 feet. If the length of the field is 28 feet more than the width, what is the length of the field (in feet)?___

Q. Ravi has $\$200$ with which to purchase bags and goggles. Each bag costs the same amount of money and each goggles cost the same amount of money. If Ravi spends all his money entirely on bags, he will get 40 bags. If he purchases only goggles, he will get 80 goggles. Which of the following best models the relationship between the number of bags $b$ and the number of goggles, $g$, that Ravi could purchase with his money?

1. $40b+80g=400$
2. $80b+40g=200$
3. $5b+2.5g=200$
4. $2.5g+5g=200$

Q. Rachel and Monica are walking. Rachel is 10 steps ahead of monica. Rachel takes one step every second while Monica takes two steps every second. After how many seconds will Monica catch up with Rachel?

1. 5
2. 10
3. 15
4. 20

Q. Two schools $A$ and $B$ want to award their selected students on the values of sincerity, truthfulness, and helpfulness. The school $A$ wants to award Rs. $x$ each Rs. $y$ each and Rs. $z$ each for the three respective values to $3,2$ and $1$ students respectively with total award money of Rs. $1600$. School $B$ wants to spend Rs. $2,300$ to award its $4,1$ and $3$ students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is Rs. $900$, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.