Chapter 4 Expressions using Letter-Numbers (Class 7 - Latest Maths NCERT (Ganita Prakash I) NCERT Solutions)

Question. Ketaki prepares and supplies coconut-jaggery laddus. The price of a coconut is ₹$35$ and the price of $1$ kg jaggery is ₹$60$. How much should she pay if she buys $8$ coconuts and $9$ kg jaggery?

Question 1. Write formulas for the perimeter of:

(a) triangle with all sides equal.

(b) a regular pentagon (as we have learnt last year, we use the word ‘regular’ to say that all sidelengths and angle measures are equal)

(c) a regular hexagon

Question 2. Munirathna has a $20$ m long pipe. However, he wants a longer watering pipe for his garden. He joins another pipe of some length to this one. Give the expression for the combined length of the pipe. Use the letter-number ‘$k$’ to denote the length in meters of the other pipe.

Question 3. What is the total amount Krithika has, if she has the following numbers of notes of ₹$100$, ₹$20$ and ₹$5$? Complete the following table:

Question 4. Venkatalakshmi owns a flour mill. It takes $10$ seconds for the roller mill to start running. Once it is running, each kg of grain takes $8$ seconds to grind into powder. Which of the expressions below describes the time taken to complete grind ‘$y$’ kg of grain, assuming the machine is off initially?

(a) $10 + 8 + y$

(b) $(10 + 8) \times y$

(c) $10 \times 8 \times y$

(d) $10 + 8 \times y$

(e) $10 \times y + 8$

Question 5. Write algebraic expressions using letters of your choice.

(a) $5$ more than a number

(b) $4$ less than a number

(c) $2$ less than $13$ times a number

(d) $13$ less than $2$ times a number

Question 6. Describe situations corresponding to the following algebraic expressions:

(a) $8 \times x + 3 \times y$

(b) $15 \times j - 2 \times k$

Question 7. In a calendar month, if any $2 \times 3$ grid full of dates is chosen as shown in the picture, write expressions for the dates in the blank cells if the bottom middle cell has date ‘$w$’.

Question. Fill the blanks below by replacing the letter-numbers by numbers; an example is shown. Then compare the values that $5u$ and $5 + u$ take.

Question. Let us compare the values that these expressions take for different values of $y$.

After filling in the two diagrams, do you think the two expressions are equal?

Question 1. Add the numbers in each picture below. Write their corresponding expressions and simplify them. Try adding the numbers in each picture in a couple different ways and see that you get the same thing.

Question 2. Simplify each of the following expressions:

(a) $p + p + p + p$, $p + p + p + q$, $p + q + p - q$

(b) $p - q + p - q$, $p + q - p + q$

(c) $p + q - (p + q)$, $p - q - p - q$

(d) $2d - d - d - d$, $2d - d - d - c$

(e) $2d - d - (d - c)$, $2d - (d - d) - c$

(f) $2d - d - c - c$

Question. Some simplifications of algebraic expressions are done below. The expression on the right-hand side should be in its simplest form.

• Observe each of them and see if there is a mistake.

• If you think there is a mistake, try to explain what might have gone wrong.

• Then, simplify it correctly.

Take a look at all the corrected simplest forms (i.e. brackets are removed, like terms are added, and terms with only numbers are also added). Is there any relation between the number of terms and the number of letter-numbers these expressions have?

Question. Find the formulas of the number machines below and write the expression for each set of inputs.

Question. Consider a set of numbers from the calendar (having endless rows) forming under the following shape:

Find the sum of all the numbers. Compare it with the number in the centre: $15$. Repeat this for another set of numbers that forms this shape. What do you observe?

We see that the total sum is always $5$ times the number in the centre. Will this always happen? How do you show this?

[Hint: Consider a general set of numbers that forms this shape. Take the number at the centre to be ‘$a$’. Express the other numbers in terms of ‘$a$’.]

Question. Look at the picture below. It is a pattern using matchsticks. Can you identify what the pattern is?

We can see that Step $1$ has $1$ triangle, Step $2$ has $2$ triangles, Step $3$ has $3$ triangles, and so on.

Matchsticks are placed in two orientations — (a) horizontal ones at the top and bottom, and (b) the ones placed diagonally in the middle.

For example, in step $2$ there are $2$ matchsticks placed horizontally and $3$ matchsticks placed diagonally.

What are these numbers in Step $3$ and Step $4$?

How does the number of matchsticks change in each orientation as the steps increase? Write an expression for the number of matchsticks at Step ‘$y$’ in each orientation. Do the two expressions add up to $2y + 1$?

Question 1. One plate of Jowar roti costs ₹$30$ and one plate of Pulao costs ₹$20$. If $x$ plates of Jowar roti and $y$ plates of pulao were ordered in a day, which expression(s) describe the total amount in rupees earned that day?

(a) $30x + 20y$

(b) $(30 + 20) \times (x + y)$

(c) $20x + 30y$

(d) $(30 + 20) \times x + y$

(e) $30x - 20y$

Question 2. Pushpita sells two types of flowers on Independence day: champak and marigold. ‘$p$’ customers only bought champak, ‘$q$’ customers only bought marigold, and ‘$r$’ customers bought both. On the same day, she gave away a tiny national flag to every customer. How many flags did she give away that day?

(a) $p + q + r$

(b) $p + q + 2r$

(c) $2 \times (p + q + r)$

(d) $p + q + r + 2$

(e) $p + q + r + 1$

(f) $2 \times (p + q)$

Question 3. A snail is trying to climb along the wall of a deep well. During the day it climbs up ‘$u$’ cm and during the night it slowly slips down ‘$d$’ cm. This happens for $10$ days and $10$ nights.

(a) Write an expression describing how far away the snail is from its starting position.

(b) What can we say about the snail’s movement if $d > u$?

Question 4. Radha is preparing for a cycling race and practices daily. The first week she cycles $5$ km every day. Every week she increases the daily distance cycled by ‘$z$’ km. How many kilometers would Radha have cycled after $3$ weeks?

Question 5. In the following figure, observe how the expression $w + 2$ becomes $4w + 20$ along one path. Fill in the missing blanks on the remaining paths. The ovals contain expressions and the boxes contain operations.

Question 6. A local train from Yahapur to Vahapur stops at three stations at equal distances along the way. The time taken in minutes to travel from one station to the next station is the same and is denoted by $t$. The train stops for $2$ minutes at each of the three stations.

(a) If $t = 4$, what is the time taken to travel from Yahapur to Vahapur?

(b) What is the algebraic expression for the time taken to travel from Yahapur to Vahapur? [Hint: Draw a rough diagram to visualise the situation]

Question 7. Simplify the following expressions:

(a) $3a + 9b - 6 + 8a - 4b - 7a + 16$

(b) $3(3a - 3b) - 8a - 4b - 16$

(c) $2(2x - 3) + 8x + 12$

(d) $8x - (2x - 3) + 12$

(e) $8h - (5 + 7h) + 9$

(f) $23 + 4(6m - 3n) - 8n - 3m - 18$

Question 8. Add the expressions given below:

(a) $4d - 7c + 9$ and $8c - 11 + 9d$

(b) $-6f + 19 - 8s$ and $-23 + 13f + 12s$

(c) $8d - 14c + 9$ and $16c - (11 + 9d)$

(d) $6f - 20 + 8s$ and $23 - 13f - 12s$

(e) $13m - 12n$ and $12n - 13m$

(f) $-26m + 24n$ and $26m - 24n$

Question 9. Subtract the expressions given below:

(a) $9a - 6b + 14$ from $6a + 9b - 18$

(b) $-15x + 13 - 9y$ from $7y - 10 + 3x$

(c) $17g + 9 - 7h$ from $11 - 10g + 3h$

(d) $9a - 6b + 14$ from $6a - (9b + 18)$

(e) $10x + 2 + 10y$ from $-3y + 8 - 3x$

(f) $8g + 4h - 10$ from $7h - 8g + 20$

Question 10. Describe situations corresponding to the following algebraic expressions:

(a) $8x + 3y$

(b) $15x - 2x$

Question 11. Imagine a straight rope. If it is cut once as shown in the picture, we get $2$ pieces. If the rope is folded once and then cut as shown, we get $3$ pieces. Observe the pattern and find the number of pieces if the rope is folded $10$ times and cut. What is the expression for the number of pieces when the rope is folded $r$ times and cut?

Question 12. Look at the matchstick pattern below. Observe and identify the pattern. How many matchsticks are required to make $10$ such squares. How many are required to make $w$ squares?

Question 13. Have you noticed how the colours change in a traffic signal? The sequence of colour changes is shown below. Find the colour at positions $90$, $190$, and $343$. Write expressions to describe the positions for each colour.

Question 14. Observe the pattern below. How many squares will be there in Step $4$, Step $10$, Step $50$? Write a general formula. How would the formula change if we want to count the number of vertices of all the squares?

Question 15. Numbers are written in a particular sequence in this endless $4$-column grid.

(a) Give expressions to generate all the numbers in a given column ($1, 2, 3, 4$).

(b) In which row and column will the following numbers appear:

(i) $124$

(ii) $147$

(iii) $201$

(c) What number appears in row $r$ and column $c$?

(d) Observe the positions of multiples of $3$. Do you see any pattern in it? List other patterns that you see.