Finding the UnknownClass 7 Mathematics NCERT Solutions

Let the blank be represented by x.

(a) $5 \times x - 8 = 37$ Add 8 to both sides: $5x = 37 + 8$ $5x = 45$ Divide by 5: $x = 9$ The integer is 9.

(b) $37 - (33 - x) = 35$ Simplify the left side: $37 - 33 + x = 35$ $4 + x = 35$ Subtract 4 from both sides: $x = 35 - 4$ $x = 31$ The integer is 31.

(c) $-3 \times (-11 + x) = 45$ Divide both sides by -3: $-11 + x = \frac{45}{-3}$ $-11 + x = -15$ Add 11 to both sides: $x = -15 + 11$ $x = -4$ The integer is -4.

Let n be the number of 50 rupee notes. Since she gets an equal number of 100 rupee notes, the number of 100 rupee notes is also n.

The total value from 50 rupee notes is 50n. The total value from 100 rupee notes is 100n.

Her total earnings are ₹ 750. We can set up an equation: (Value from 50 rupee notes) + (Value from 100 rupee notes) = Total earnings $50n + 100n = 750$

Now, we solve for n: $150n = 750$ $n = \frac{750}{150}$ $n = 5$

Therefore, Ranju has 5 notes of each denomination (five 50 rupee notes and five 100 rupee notes).

From the picture, we can see:

Let x be the number of dots hidden under one black blob.

The total number of hidden dots is the number of blobs multiplied by the number of dots under each blob, which is 5x.

The total number of dots in the picture is the sum of the hidden dots and the visible dots.

Equation: Total dots = (Number of blobs × dots per blob) + visible dots $25 = 5x + 5$

Therefore, there are 4 dots covered by one blob.

(a) Machine: Multiply by 3, then Add 5 Let the input be x and the output be y. The equation for this machine is $y = 3x + 5$. To find the input x, we can rearrange the equation: $x = \frac{y-5}{3}$.

(b) Machine: Subtract 4, then Divide by 5 Let the input be x and the output be y. The equation for this machine is $y = \frac{x-4}{5}$. To find the input x, we can rearrange the equation: $x = 5y + 4$.

Let the initial input be x. We will trace the operations step by step.

The final output is 20. So, we can set up the equation: $\frac{5(x + 4) - 10}{2} = 20$

Now, we solve for x:

Therefore, the initial input is 6.

Let k be the number of kilometres traveled.

The cost is calculated as: Total Cost = Fixed Fee + (Cost per km × Number of km)

We are given:

We can form the equation: $2200 = 800 + 20k$

Now, we solve for k:

Therefore, the number of kilometres traveled is 70.

Let one number be x. Since the other number is three times the first, the second number is 3x.

The sum of the two numbers is 76. We can write the equation: $x + 3x = 76$

Now, we solve for x: $4x = 76$ $x = \frac{76}{4}$ $x = 19$

The first number is 19. The second number is $3x = 3(19) = 57$.

The two numbers are 19 and 57.

Check: $19 + 57 = 76$. The solution is correct.

Based on the figure provided in the textbook:

Let g be the width of one gap.

First, we find the internal width of the window available for the grill (rods and gaps) by subtracting the frame width from both sides: Internal width = Total width - (2 × Frame width) Internal width = $100 - (2 \times 5) = 100 - 10 = 90$ cm.

Next, we calculate the total width taken up by the 4 rods: Total rod width = Number of rods × Width of one rod Total rod width = $4 \times 10 = 40$ cm.

The internal width is the sum of the total width of the rods and the total width of the gaps. Internal width = Total rod width + Total gap width $90 = 40 + (5 \times g)$

Now, we solve for g:

Therefore, the gap between two rods is 10 cm.

Let the cost of a chocolate milkshake be m rupees. Since a fruit juice costs ₹15 less than a milkshake, the cost of a fruit juice is m - 15 rupees.

The total cost of 4 fruit juices and 7 chocolate milkshakes is ₹600. We can form an equation: $4 \times ( ext{cost of fruit juice}) + 7 \times ( ext{cost of milkshake}) = 600$ $4(m - 15) + 7m = 600$

Now, we solve for m:

So, the cost of a chocolate milkshake is ₹60.

The cost of a fruit juice is $m - 15 = 60 - 15 = 45$.

Therefore, the cost of a fruit juice is ₹45 and the cost of a chocolate milkshake is ₹60.

We are given the equation $28p - 36 = 98$.

To find the value of $14p - 19$: Notice that the terms in $14p - 19$ are half of the terms $28p$ and close to half of $36$. Let's manipulate the original equation. $28p - 36 = 98$ Divide every term in the equation by 2: $\frac{28p}{2} - \frac{36}{2} = \frac{98}{2}$ $14p - 18 = 49$ We need to find $14p - 19$. We can rewrite this as $(14p - 18) - 1$. Since $14p - 18 = 49$, the value is: $49 - 1 = 48$. So, $14p - 19 = 48$.

To find the value of $28p - 38$: We start with the given equation: $28p - 36 = 98$. We want to find the value of $28p - 38$. We can rewrite this expression as $(28p - 36) - 2$. Since we know $28p - 36 = 98$, we can substitute this value: $98 - 2 = 96$. So, $28p - 38 = 96$.

The question requires figures of triangles with angles expressed in terms of variables. As these figures are not provided in the source content, this question cannot be answered.

We can generate equations by starting with the solution $u=6$ and applying the same operation to both sides.

Let the amount given to the first person be x.

The total amount distributed is 132. We can form the equation by summing the amounts: $x + 2x + 6x + 24x = 132$

Now, we solve for x: $33x = 132$ $x = \frac{132}{33}$ $x = 4$

Therefore, the amount given to the first person is 4.

Let the height of the giraffe be h metres. Two and a half metres can be written as 2.5 metres.

According to the problem, we can form the equation: Height = (Half its height) + 2.5 metres $h = \frac{h}{2} + 2.5$

To solve for h, we subtract $\frac{h}{2}$ from both sides: $h - \frac{h}{2} = 2.5$ $\frac{h}{2} = 2.5$

Now, multiply both sides by 2: $h = 2.5 \times 2$ $h = 5$

Therefore, the giraffe is 5 metres tall.

Based on the figures in the textbook, there are two sequences.

(a) In position 11, the number of squares is 11.

(b) For position 11, the number of sticks is $3(11) + 1 = 33 + 1 = **34**$.

(c) To check for 85 sticks, we set up the equation: $3n + 1 = 85$. $3n = 84$ $n = 28$ Yes, an arrangement can be made, and it will be at position 28.

(d) To check for 150 sticks, we set up the equation: $3n + 1 = 150$. $3n = 149$ $n = \frac{149}{3} = 49.66...$ Since n must be a whole number, no, an arrangement cannot be made with exactly 150 sticks.

(a) In position 11, the number of triangles is 11.

(b) For position 11, the number of sticks is $2(11) + 1 = 22 + 1 = **23**$.

(c) To check for 85 sticks, we set up the equation: $2n + 1 = 85$. $2n = 84$ $n = 42$ Yes, an arrangement can be made, and it will be at position 42.

(d) To check for 150 sticks, we set up the equation: $2n + 1 = 150$. $2n = 149$ $n = \frac{149}{2} = 74.5$ Since n must be a whole number, no, an arrangement cannot be made with exactly 150 sticks.

Let the number be x.

According to the problem, we can form the equation: Number + 36 = 10 × Number $x + 36 = 10x$

Now, we solve for x:

Therefore, the number is 4.

(a) $5(r+2)=10$ Divide by 5: $r+2 = 2$. Subtract 2: $r = 0$.

(b) $-3(u+2)=2(u-1)$ Distribute: $-3u - 6 = 2u - 2$. Add 3u to both sides: $-6 = 5u - 2$. Add 2 to both sides: $-4 = 5u$. Divide by 5: $u = -\frac{4}{5}$.

(c) $2(7-2n)=-6$ Divide by 2: $7 - 2n = -3$. Subtract 7: $-2n = -10$. Divide by -2: $n = 5$.

(d) $2(x-4)=-16$ Divide by 2: $x - 4 = -8$. Add 4: $x = -4$.

(e) $6(x-1)=2(x-1)-4$ Distribute: $6x - 6 = 2x - 2 - 4$. Simplify: $6x - 6 = 2x - 6$. Subtract 2x: $4x - 6 = -6$. Add 6: $4x = 0$. Divide by 4: $x = 0$.

(f) $3-7s=7-3s$ Add 7s to both sides: $3 = 7 + 4s$. Subtract 7: $-4 = 4s$. Divide by 4: $s = -1$.

(g) $2x+1=6-(2x-3)$ Simplify RHS: $2x+1 = 6 - 2x + 3$. Simplify RHS: $2x+1 = 9 - 2x$. Add 2x to both sides: $4x + 1 = 9$. Subtract 1: $4x = 8$. Divide by 4: $x = 2$.

(h) $10-5x=3(x-4)-2(x-7)$ Distribute on RHS: $10-5x = 3x - 12 - 2x + 14$. Simplify RHS: $10-5x = x + 2$. Add 5x to both sides: $10 = 6x + 2$. Subtract 2: $8 = 6x$. Divide by 6: $x = \frac{8}{6} = \frac{4}{3}$.

To find the path, we must solve the equation in each box and move to the next box whose solution matches the variable's value from the current box.

The correct path is: START $\rightarrow$ ($x-3=-5$) $\rightarrow$ ($3x+2=x-2$) $\rightarrow$ ($5x-3=3x-7$) $\rightarrow$ ($3(x+1)=x-1$) $\rightarrow$ ($\frac{x}{2}+5=4$) $\rightarrow$ END.

Let c be the number of children and d be the number of donkeys.

Each child and each donkey has 1 head. The total number of heads is 28. Equation 1: $c + d = 28$

Each child has 2 feet and each donkey has 4 feet. The total number of feet is 80. Equation 2: $2c + 4d = 80$

We can solve this system of equations. From Equation 1, we can express c in terms of d: $c = 28 - d$

Now substitute this expression for c into Equation 2: $2(28 - d) + 4d = 80$

Solve for d:

There are 12 donkeys.

Now find the number of children using $c = 28 - d$: $c = 28 - 12$ $c = 16$

There are 16 children.

Check: Heads: $16 + 12 = 28$. Correct. Feet: $(2 \times 16) + (4 \times 12) = 32 + 48 = 80$. Correct.

To create equations with a solution of $x=-2$, we can start with the solution and perform the same operation on both sides.

(a) $2y = 60$ Divide both sides by 2: $\frac{2y}{2} = \frac{60}{2}$ $y = 30$

(b) $-8 = 5x - 3$ Add 3 to both sides: $-8 + 3 = 5x - 3 + 3$ $-5 = 5x$ Divide both sides by 5: $\frac{-5}{5} = \frac{5x}{5}$ $x = -1$

(c) $-53w = -15$ Divide both sides by -53: $\frac{-53w}{-53} = \frac{-15}{-53}$ $w = \frac{15}{53}$

(d) $13 - z = 8$ Subtract 13 from both sides: $13 - 13 - z = 8 - 13$ $-z = -5$ Multiply both sides by -1: $z = 5$

(e) $k + 8 = 12 - k$ Add k to both sides: $k + k + 8 = 12 - k + k$ $2k + 8 = 12$ Subtract 8 from both sides: $2k + 8 - 8 = 12 - 8$ $2k = 4$ Divide both sides by 2: $\frac{2k}{2} = \frac{4}{2}$ $k = 2$

(f) $7m = m - 3$ Subtract m from both sides: $7m - m = m - m - 3$ $6m = -3$ Divide both sides by 6: $\frac{6m}{6} = \frac{-3}{6}$ $m = -\frac{1}{2}$

(g) $3n = 10 + n$ Subtract n from both sides: $3n - n = 10 + n - n$ $2n = 10$ Divide both sides by 2: $\frac{2n}{2} = \frac{10}{2}$ $n = 5$

Let the ten's digit be t. According to the problem:

The sum of the three digits is 15. We can form an equation: (Hundred's digit) + (Ten's digit) + (Unit's digit) = 15 $(t - 3) + t + (t + 3) = 15$

Now, we solve for t: $3t - 3 + 3 = 15$ $3t = 15$ $t = \frac{15}{3}$ $t = 5$

So, the ten's digit is 5.

The number is formed by the digits 2, 5, and 8 in the hundreds, tens, and units place respectively.

Therefore, the number is 258.

Let the weight of the brick be w kg. According to the problem, we can write the following equation: Weight of the brick = (Half its weight) + 1 kg $w = \frac{w}{2} + 1$

To solve for w, we first subtract $\frac{w}{2}$ from both sides: $w - \frac{w}{2} = \frac{w}{2} - \frac{w}{2} + 1$ $\frac{w}{2} = 1$

Now, multiply both sides by 2: $\frac{w}{2} \times 2 = 1 \times 2$ $w = 2$

Therefore, the weight of the brick is 2 kg.

Let the number be n. According to the problem, we can form the equation: (One quarter of the number) + 9 = The number $\frac{1}{4}n + 9 = n$

To solve for n, we subtract $\frac{1}{4}n$ from both sides: $9 = n - \frac{1}{4}n$ $9 = \frac{3}{4}n$

Now, we can find n by multiplying both sides by $\frac{4}{3}$: $9 \times \frac{4}{3} = \frac{3}{4}n \times \frac{4}{3}$ $3 \times 4 = n$ $n = 12$

Therefore, the number is 12.

First, we solve the given equation $4k + 1 = 13$ to find the value of k. $4k + 1 = 13$ Subtract 1 from both sides: $4k = 13 - 1$ $4k = 12$ Divide by 4: $k = \frac{12}{4}$ $k = 3$

Now we can find the values of the given expressions.

(a) $8k + 2$ Substitute $k=3$: $8(3) + 2 = 24 + 2 = 26$ Alternatively, notice that $8k+2$ is $2(4k+1)$. Since $4k+1=13$, the value is $2(13) = 26$.

(b) $4k$ From our initial calculation, we found $4k = 12$.

(c) $k$ From our initial calculation, we found $k = 3$.

(d) $4k - 1$ We know $4k = 12$. So, $4k - 1 = 12 - 1 = 11$.

(e) $-k - 2$ Substitute $k=3$: $-(3) - 2 = -3 - 2 = -5$

An equation has no solution if the statement of equality is false, regardless of the value of the variable.

Following the hint, let the number be x.

For these to be equal, we frame the equation: $x + 4 = x + 5$

To solve this, we can subtract x from both sides: $x - x + 4 = x - x + 5$ $4 = 5$

This statement, $4 = 5$, is always false. Therefore, there is no value of x that can make the original equation true. The equation $x + 4 = x + 5$ has no solution.

Am I correct? Yes, the prediction is correct. The final number will always be 8, regardless of the starting number.

Can you explain why the trick works? Yes, we can use algebra to explain it. Let the number you think of be x.

As you can see, the variable x is eliminated in step 5. This means the result of the first five steps is always 5, no matter what number you start with. Adding 3 in the final step always results in 8.

Can you make your own such tricks? Yes. The key is to perform a series of operations that eventually cancel out the original unknown number (x).

Here is an example of a new trick: