Finding the Unknown

IT'S PUZZLE TIME!

A Magic Trick

Think of any number. Now multiply it by 2. Add 10. Divide by 2. Now subtract the original number you thought of. Finally, add 3.

I predict that you now have 8. Am I correct? Try the trick on your friends and family! Can you explain why the trick works? [Hint: Denote the first number thought of by x.] Can you make your own such tricks?

Solve these equations and check the solutions.

(a) $3x - 10 = 35$ (b) $5s = 3s$ (c) $3u - 7 = 2u + 3$ (d) $4(m + 6) - 8 = 2m - 4$ (e) $\frac{u}{15} = 6$

Frame an equation that has no solution.

[Hint: 4 more than a number, and 5 more than a number can never be equal!]

Write 5 equations whose solution is $x = -2$.

Find the value of each unknown:

(a) $2y = 60$ (b) $-8 = 5x - 3$ (c) $-53w = -15$ (d) $13 - z = 8$ (e) $k + 8 = 12 - k$ (f) $7m = m - 3$ (g) $3n = 10 + n$

I am a 3-digit number. My hundred's digit is 3 less than my ten's digit. My ten's digit is 3 less than my unit's digit. The sum of all the three digits is 15. Who am I?

The weight of a brick is 1 kg more than half its weight. What is the weight of the brick?

One quarter of a number increased by 9 gives the same number. What is the number?

Given $4k + 1 = 13$, find the values of:

(a) $8k + 2$ (b) $4k$ (c) $k$ (d) $4k - 1$ (e) $-k - 2$

Fill in the blanks with integers.

(a) $5 \times \underline{\quad} - 8 = 37$

(b) $37 - (33 - \underline{\quad}) = 35$

(c) $-3 \times (-11 + \underline{\quad}) = 45$

Ranju is a daily wage labourer. She earns ₹ 750 a day. Her employer pays her in 50 and 100 rupee notes. If Ranju gets an equal number of 50 and 100 rupee notes, how many notes of each does she have?

In the given picture, each black blob hides an equal number of blue dots. If there are 25 dots in total, how many dots are covered by one blob? Write an equation to describe this problem.

Here are machines that take an input, perform an operation on it and send out the result as an output.

What are the inputs to these machines?

A taxi driver charges a fixed fee of ₹800 per day plus ₹20 for each kilometer traveled. If the total cost for a taxi ride is ₹2200, determine the number of kilometres traveled.

The sum of two numbers is 76. One number is three times the other number. What are the numbers?

The figure shows the diagram for a window with a grill. What is the gap between two rods in the grill?

In a restaurant, a fruit juice costs ₹15 less than a chocolate milkshake. If 4 fruit juices and 7 chocolate milkshakes cost ₹600, find the cost of the fruit juice and milkshake.

Given $28p - 36 = 98$, find the value of $14p - 19$ and $28p - 38$.

The steps to solve three equations are shown below. Identify and correct any mistakes.

(a) $6x + 9 = 66$

$x + 9 = 11$

$x = 11 - 9$

$x = 2$

(b) $14y + 24 = 36$

$7y + 12 = 18$

$7y = 6$

$y = \frac{6}{7}$

(c) $4x - 5 = 9x + 8$

$4x = 9x + 8 - 5$

$4x = 9x + 3$

$4x - 9x = 3$

$-5x = 3$

$x = \frac{-5}{3}$

Find the measures of the angles of these triangles.

Write 4 equations whose solution is $u = 6$.

The Bakhshālī Manuscript (300 CE) mentions the following problem. The amount given to the first person is not known. The second person is given twice as much as the first. The third person is given thrice as much as the second; and the fourth person four times as much as the third. The total amount distributed is 132. What is the amount given to the first person?

The height of a giraffe is two and a half metres more than half its height. How tall is the giraffe?

Two separate figures are given below. Each figure shows the first few positions in a sequence of arrangements made with sticks. Identify the pattern and answer the following questions for each figure:

(a) How many squares are in position number 11 of the sequence? (b) How many sticks are needed to make the arrangement in position number 11 of the sequence? (c) Can an arrangement in this sequence be made using exactly 85 sticks? If yes, which position number will it correspond to? (d) Can an arrangement in this sequence be made using exactly 150 sticks? If yes, which position number will it correspond to?

A number increased by 36 is equal to ten times itself. What is the number?

Solve these equations:

(a) $5(r + 2) = 10$

(b) $-3(u + 2) = 2(u - 1)$

(c) $2(7 - 2n) = -6$

(d) $2(x - 4) = -16$

(e) $6(x - 1) = 2(x - 1) - 4$

(f) $3 - 7s = 7 - 3s$

(g) $2x + 1 = 6 - (2x - 3)$

(h) $10 - 5x = 3(x - 4) - 2(x - 7)$

Solve the equations to find a path from Start to the End. Show your work in the given boxes provided and colour your path as you proceed.

Find the unknown weights in the following cases:

Context: Finding the Unknown

Q. Discuss the answers with your classmates. Give reasons why you think your answer is right.

Find the unknown weight of the sack in the following cases. In Fig. 7.10, all the sacks have the same weight.

[Hint: If we remove equal weights from both the plates, will the weighing scale still be balanced? Remove one sack from each plate for Fig. 7.10.]

Jasmine decides to make a matchstick arrangement that appears in this sequence, using exactly 99 sticks. What will be the position number of this arrangement in the sequence?

Can you find ways to get the value of n, such that 2n + 1 = 99?

Is it possible to make a matchstick arrangement that appears in this sequence using exactly 200 sticks?

For the weighing scale problems in figures 7.6, 7.7, 7.8, 7.9, 7.10, and 7.11, frame equations by using letter-numbers to denote the unknown weight.

Context: For the problem in Fig. 7.6, let us denote the weight of one fried egg as $e$. Since each slice of bread is 2, we have $2 + 2 + 2 = 6$ on one side and $e + e$ on the other side. Since they are equal, we have $6 = e + e$, or $2e = 6$.

For the problem in Fig. 7.7, $\text{star} = 4$, and we can denote the weight of one $\text{ring}$ as $y$. So, we have 16 on one side, and $4 + 2y$ on the other side. Thus, we have the equation $4 + 2y = 16$.

Q. Solve the equations that you frame and check if you get the same value for the unknown weight as you got previously.

Context: For the problem in Fig. 7.6, let us denote the weight of one fried egg as $e$. Since each slice of bread is $2$, we have $2 + 2 + 2 = 6$ on one side and $e + e$ on the other side. Since they are equal, we have: $6 = e + e\text{, or}$ $2e = 6$

For the problem in Fig. 7.7, [flower] $= 4$, and we can denote the weight of one [donut] as $y$. So, we have $16$ on one side, and $4 + 2y$ on the other side. Thus, we have the equation: $4 + 2y = 16$

Q. Frame 5 equations. Find methods to solve them.

Context: Consider the equation $2n + 1 = 99$.

Q. Can this equation have any other solution?

Try solving $5x - 4 = 7$ using trial and error.

Consider an equation $15 + 8 = 23$. If we add, subtract, multiply or divide the same number on both sides, will it still preserve the equality of LHS and RHS?

Context: It is known that $23 \times 41 \times 11 \times 8 \times 7 = 5,80,888$. To find the value of the expression $23 \times 41 \times 11 \times 8$, we can divide $5,80,888$ by $7$.

Q. Is this the same as dividing both sides by 7, which removes the factor 7 and leaves only the expression to be evaluated on the LHS?

Ranjana creates a sequence of arrangements with square tiles as shown below. Can she extend the sequence and make an arrangement using 100 tiles? If yes, which step in the sequence will it be?

We have the expression $3k + 1$ which gives the number of tiles needed to make an arrangement in Step $k$. To check whether an arrangement is possible using 100 tiles at some Step $k$, we can solve the equation: $3k + 1 = 100$. Find the value of $k$.

We have the expression $3k + 1$ which gives the number of tiles needed to make an arrangement in Step $k$.

To check whether an arrangement is possible using 100 tiles at some Step $k$, we can solve the equation: $3k + 1 = 100$. Find the value of $k$.

Context: Example 11: Riyaz created a math trick, which he tries out on his friend Akash.

Riyaz asked Akash to perform the following steps without revealing the answer to any of the intermediate steps.

1. Think of a number.
2. Subtract 3 from the number.
3. Multiply the result by 4.
4. Add 8 to the product.
5. Reveal the final answer.

The final answer revealed by Akash was 24. Using this, Riyaz correctly figured out the starting number that Akash had thought of. Find this number.

Q. Try the steps using different numbers as the starting number. Do you see any relation between the starting number and final answer?

Context: Since Akash's final answer was 24, we have the equation: $4x - 4 = 24$ $4(x - 1) = 24$ $x - 1 = 6 \quad \text{(Dividing both sides by 4)}$ Thus, Akash thought of the number 7.

Q. Can you think of a simple rule that you can use to get the starting number from the final answer?

Context: Ramesh and Suresh have 60 marbles between them. Ramesh has 30 more marbles than Suresh. If the number of marbles with Suresh is $y$, then the number of marbles with Ramesh is $y + 30$. Since the total number of marbles is 60, we have the equation: $2y + 30 = 60$

Q. Use this to find both the unknowns.

Write equations whose solution is $y = 5$. Share the equations you made with each other and discuss the methods used.

Can you form a chain going from the bottom equation to the top? Compare the operations used when going from the top to the bottom and from the bottom to the top.

Without calculating, can you find the value of the unknown in each equation in the chains above?

[Hint: We have seen that the value that satisfies an equation also satisfies the new equation obtained by performing the same operation on both sides of the original equation.]

7.3 Mind the Mistake, Mend the Mistake

The following are some equations along with the steps used to solve them to find the value of the letter-number. Go through each solution and decide whether the steps are correct. If there is a mistake, describe the mistake, correct it and solve the equation.

Context: Consider the equations $5x + 4 = 3x + 8$ and $3x - 6 = 2x + 4$.

Q. Can we come up with a formula to solve these equations? That is, for the first equation, can we perform some operations using 5, 4, 3, and 8 that will directly give us the solution? Using a similar method, can you solve the second equation using the numbers 3, – 6, 2 and 4?

Context: Brahmagupta's formula for solving equations of the form $Ax + B = Cx + D$ is $x = \frac{D - B}{A - C}$.

Q. Using this formula can you solve this equation $2x + 3 = 4x + 5$?

There are some children and donkeys on a beach. Together they have 28 heads and 80 feet. How many donkeys are there? How many children are there?