Arithmetic Expressions | CBSE Class 7 Maths Notes

This note covers arithmetic expressions and their values, comparison by reasoning, brackets, signed terms, swapping and grouping terms, removing brackets, the distributive property, efficient calculation, and expressions for everyday situations.

What is an arithmetic expression and how is it read?

Definition: An arithmetic expression is a mathematical phrase made using numbers and arithmetic operations. Its value is the number obtained when those operations are carried out.

An operation is an action on numbers: addition, subtraction, multiplication or division. The symbols +, −, × and ÷ mean add, subtract, multiply and divide, respectively. The equality sign = means that the quantities on its two sides have the same value.

For example, 13 + 2 is an expression, while 15 is its value. Writing 13 + 2 = 15 connects the expression to that value. Read 13 + 2 as “13 plus 2” or “the sum of 13 and 2”. A sum is the result of addition. A difference is the result of subtraction.

How do words describe multiplication?

The expression 5 × 25 means five times twenty-five. The result of multiplying numbers is their product. Thus, “the product of 5 and 25” describes the same expression. The order of the words should make clear which operation is being described.

Worked example 1. Mallika spends ₹25 on school lunch each day from Monday to Friday. Here ₹ means rupees. Write an expression for her total lunch spending.

Answer: There are five lunch days, each costing ₹25. The expression is 5 × 25. Evaluating it gives ₹125 for the five days.

Can different expressions represent one value?

Different calculations can finish at the same number. The expressions in this table all represent 12, even though they use different operations. Equality concerns their values, rather than whether their written forms look alike.

Keep the distinction between an expression and its value in mind when translating a situation. The expression shows how the numbers are connected. Evaluating it gives the final number, which must then be interpreted in the situation, such as an amount of money.

How can expressions be compared without lengthy calculation?

Expressions are compared by their values. The sign > means greater than, and < means less than. For example, 10 + 2 > 7 + 1 because the values are 12 and 8. Similarly, 13 − 2 < 4 × 3.

It is sometimes possible to compare expressions by examining what changes between them. This avoids calculating both complete values. With addition, consider how much each number increases or decreases. With subtraction, consider changes in both the starting amount and the amount removed.

What happens when two additions change?

Worked example 2. Compare 1023 + 125 and 1022 + 128 without calculating both totals.

Answer: Imagine Raja beginning with 1023 marbles and receiving 125, while Joy begins with 1022 and receives 128. Raja begins with one more marble, but Joy receives three more. Joy therefore finishes with two more: 1023 + 125 < 1022 + 128.

Yellow bars labelled 1022 and 125 show matching amounts for Raja and Joy. Raja has one extra circle labelled 1, while Joy has three. These extra circles show why Joy's total is larger by two.

Reference: NCERT Class 7, page 25, unnumbered diagram

Can two subtractions give equal results?

Compare 113 − 25 and 112 − 24. Raja begins with one more marble than Joy but also loses one more. The extra starting marble is removed, so the amounts left are equal: 113 − 25 = 112 − 24.

This reasoning is useful when the numbers are close. Do not compare only the first number and stop. In both examples, the second part of the expression also affects the result. A larger starting amount alone does not settle which complete expression has the larger value.

Ascending order means increasing order of value. The expressions 120 ÷ 3, 67 − 20, 67 − 19, 5 × 11 and 35 + 25 are in ascending order. Their positions depend on their evaluated values, not on the sizes of their first written numbers.

Why do brackets matter in an arithmetic expression?

Brackets, written ( ), group an expression whose value must be found before carrying out the surrounding operations. They make clear which numbers and operations belong together. In a situation involving several actions, the grouped expression should represent the quantity that must be treated as a whole.

How do brackets show the intended calculation?

Worked example 3. Mallesh brings 30 marbles. Arun brings five bags containing four marbles each. Find their combined number of marbles.

Answer: Arun brings 5 × 4 marbles. Therefore, the total is 30 + (5 × 4) = 30 + 20 = 50 marbles. The brackets show that the five groups must be counted before adding Mallesh's 30.

Adding 30 and 5 first and then multiplying by 4 gives 140. That calculation fails to describe the situation: Mallesh's 30 marbles are not five bags of four and should not be multiplied by four. The meaning of the quantities explains the grouping.

Why must a total purchase cost be grouped?

Worked example 4. Irfan buys biscuits for ₹15 and toor dal for ₹56, then pays ₹100. Find his change using brackets.

Answer: His total purchase cost is 15 + 56. Subtract that whole cost from his payment: 100 − (15 + 56) = 100 − 71 = 29. Irfan receives ₹29 in change.

Change is the amount returned after the cost of a purchase is taken from the payment. Writing 100 − 15 + 56 would subtract the biscuit cost but then add the dal cost. Its value, 141, is not the required change.

These examples show why understanding a situation comes before choosing the order of calculation. A bracket is not decoration. Moving or omitting it can change the quantity being described. Read the whole grouped expression before replacing it with its value.

How do we identify and evaluate the terms of an expression?

Definition: Terms are the parts of an expression separated by addition signs after subtraction has been rewritten as addition of the inverse.

Here the inverse of a number means the number with the opposite sign: the inverse of 14 is −14, and the inverse of −14 is 14. The sign − before a number marks a negative number. Subtracting 14 is equivalent to adding −14.

Thus, 83 − 14 is written as 83 + (−14), with terms 83 and −14. A signed term is a term considered together with its positive or negative sign. The sign belongs to the term. Similarly, −18 − 3 has terms −18 and −3. A product such as 6 × 5 remains a single term; the multiplication sign does not separate terms.

How do signed terms guide evaluation?

In 2 − 10 + 4 × 6, the terms are 2, −10 and 4 × 6. Evaluate the product as part of its term before adding the terms. Grouping inside brackets must also be respected when evaluating a term.

1. Read the brackets and identify the quantities they group.
2. Rewrite each subtraction as addition of the inverse when identifying terms.
3. Evaluate each term, including the products or divisions within it.
4. Add the evaluated terms, retaining the sign attached to each term.

Worked example 5. Identify the terms of 48 − 10 × 2 + 16 ÷ 2 and evaluate the expression.

Answer: The terms are 48, −10 × 2 and 16 ÷ 2. They evaluate to 48, −20 and 8. Adding them gives 48 + (−20) + 8 = 36.

What if a term itself contains brackets?

The expression 5 × (3 + 2) + 7 × 8 + 3 has the outer terms 5 × (3 + 2), 7 × 8 and 3. Here “outer terms” means the terms of the whole expression, with the bracketed quantity kept together.

First evaluate 3 + 2 as 5, making the first term 25. The second term is 56 and the last term is 3. Their sum is 25 + 56 + 3 = 84. The plus sign inside the brackets belongs to the grouped expression, not to the separation of the outer terms.

Why can signed terms be swapped and grouped?

Property: Commutative property of addition

The commutative property of addition means that swapping two terms does not change their sum. This applies when negative terms are present as well as when both terms are positive. The whole signed term must move together.

Madhu's drone goes 6 m up and then 4 m down from a terrace, where m means metres. Its final height above the terrace is 6 − 4 = 2 m. As a sum, 6 + (−4) has the same value as (−4) + 6.

Property: Associative property of addition

The associative property of addition means that changing the grouping of terms does not change their sum. For three terms, one may add the first two and then the third, or add the last two and then the first.

Worked example 6. Explain how the terms of (−7) + 10 + (−11) can be added in different orders.

Answer: Grouping the first two terms gives 3 + (−11) = −8. Grouping the last two gives (−7) + (−1) = −8. Adding the negative terms first gives (−18) + 10 = −8. Each method preserves the signed terms.

Together, these properties explain why terms in a sum can be added in any order. They do not say that every operation can be rearranged without care. In particular, rewriting subtraction as addition makes it clear which negative quantity must remain attached to its sign.

How can this avoid repeating a calculation?

Manasa adds a list of numbers and obtains 11749, then discovers that she omitted 9055. She does not need to restart the entire addition. She can add the missing number to the total already found, giving 11749 + 9055 = 20804.

The missing number's position in the original list does not change the sum. This is a practical use of swapping and grouping terms: the completed sum can be treated as one group, and the omitted term added afterwards. The other terms do not need to be added again.

How do terms represent groups, packets and arrangements?

An expression should preserve what each number counts. Multiplication can describe equal groups, division can describe splitting a quantity into equal packets, and addition can join the resulting quantities. Before calculating, explain the meaning of every term in the situation.

How are equal groups and an extra amount combined?

Worked example 7. Four friends order four dosas costing ₹23 each and leave a total tip of ₹5. Write and evaluate the expression for their payment.

Answer: The dosa cost is 4 × 23 and the tip is 5. The two terms are therefore 4 × 23 and 5. The payment is 4 × 23 + 5 = 92 + 5 = ₹97.

A tip is the extra amount given to thank the waiter. If seven friends each order a dosa at the same price and the total tip remains ₹5, the expression becomes 7 × 23 + 5. Its value is ₹166, with terms 7 × 23 and 5.

In Ruby's game, 33 students form groups when a number is called. When the number is 5, six complete groups form and three students remain. The expression 6 × 5 + 3 records both parts. Its terms describe grouped students and the students left over.

How does division become a term?

Worked example 8. Raghu buys 100 kg of rice and packs it into 2 kg packets. He already has four such packets. Here kg means kilograms. How many packets does he now have?

Answer: The new packets number 100 ÷ 2, so the complete expression is 4 + 100 ÷ 2. Its terms are 4 and 100 ÷ 2. He has 4 + 50 = 54 packets.

The left arrangement has five green columns of two squares and one pink column of three. The right arrangement has two columns, each containing five yellow squares above three blue squares.

Reference: NCERT Class 7, page 33, unnumbered diagram

The left arrangement represents 5 × 2 + 3, whose value is 13. The right represents 2 × (5 + 3). It can also be described by 5 + 3 + 5 + 3 or by 5 × 2 + 3 × 2. The brackets group the contents of each complete column.

How are brackets removed from sums and differences?

Removing brackets means rewriting an expression without changing its value. The operation before the brackets matters. First decide whether the grouped quantity is being added or subtracted, then follow what that means for the signed terms inside.

What happens when a whole sum is subtracted?

In 200 − (40 + 3), the amount removed is 43. It can be removed by subtracting 40 and then subtracting 3. Therefore, 200 − (40 + 3) = 200 − 40 − 3 = 157. Adding 3 after subtracting 40 would not remove the whole amount.

The same reasoning applies to Irfan's change. His payment of ₹100 must cover both the ₹15 biscuits and the ₹56 dal. Subtracting the purchases separately gives 100 − 15 − 56, equal to 100 − (15 + 56). Both routes give ₹29.

What happens when a difference is subtracted?

Worked example 9. Remove the brackets in 500 − (250 − 100) and find the value.

Answer: The bracketed amount is 150. Subtracting 250 directly would remove 100 too much, so add 100 back. Thus, 500 − (250 − 100) = 500 − 250 + 100 = 350.

When removing brackets preceded by a negative sign, change the signs of the terms inside. In this example, 250 becomes −250, and −100 becomes +100. Thinking about the amount removed explains the sign change instead of leaving it as an unexplained instruction.

What happens when the grouped quantity is added?

Hira has 28 coins in one bag and 35 in another. She gives away 10 from the second bag. Her remaining coins are represented by 28 + (35 − 10). Since the remaining second bag is added to the first, the expression becomes 28 + 35 − 10 = 53.

Note: These addition and subtraction examples concern a grouped quantity being added or subtracted. When a number multiplies a bracketed expression, use the distributive property described next.

How does the distributive property explain multiplication with brackets?

Property: Distributive property

The distributive property connects multiplication with addition or subtraction. Multiplying a sum by a number gives the sum of the separate products. Multiplying a difference by a number gives the difference of the separate products. Each term inside the brackets must be multiplied.

Why must both parts of a meal be multiplied?

Worked example 10. Lhamo and Norbu each order a vegetable cutlet costing ₹43 and a rasgulla costing ₹24. Write two equal expressions for their combined bill.

Answer: One person's bill is 43 + 24. Two such bills give 2 × (43 + 24). Counting the items separately gives 2 × 43 + 2 × 24. Therefore, 2 × (43 + 24) = 2 × 43 + 2 × 24.

The expression 2 × 43 + 24 would count two cutlets but only one rasgulla. It therefore does not represent the stated meal. The bracket tells us that both food costs belong to the amount being doubled.

How can rows be combined before multiplication?

In a Republic Day parade, the scouts form four rows of five and the guides form three rows of five. Counting separately gives 4 × 5 + 3 × 5. Counting all rows first gives (4 + 3) × 5. Both count 35 children.

Three rows of five green symbols are grouped with the label 3 × 5. Four rows of five yellow symbols are grouped with the label 4 × 5. A larger brace joins both groups as (4 + 3) × 5.

Reference: NCERT Class 7, page 39, unnumbered diagram

The same reasoning gives 10 × 98 + 3 × 98 = (10 + 3) × 98. Ten lots and three lots of the same amount together make thirteen lots. This rewrites the calculation while keeping the quantity being counted unchanged.

For subtraction, 14 × 10 − 6 × 10 = (14 − 6) × 10. Removing six lots of ten from fourteen lots leaves eight lots. The number multiplying each part must remain attached to both products when the brackets are removed.

How can changing terms make calculations easier?

Knowing how an expression changes can save work. If one term in a sum increases while the others remain unchanged, the total increases by the same amount. With a product, changing the number of equal groups changes the total by that many groups.

How do signed terms affect a sum?

Start with 53 + (−16) = 37. Replacing 53 by 54 raises the result to 38. Replacing −16 by −15 also increases a term by one. A negative sign does not mean that a change must lower the sum; compare the signed values themselves.

Jasoda's method for subtracting 9 is to subtract 10 and then add 1. The expression 36 − 9 can be rewritten as 36 − (10 − 1), which becomes 36 − 10 + 1. Adding one corrects the extra amount removed.

How can a known product give a new product?

Worked example 11. Given that 53 × 18 = 954, find 63 × 18 without starting the multiplication again.

Answer: There are ten extra groups of 18. Write 63 × 18 = (53 + 10) × 18 = 53 × 18 + 10 × 18. Substitute the known product to obtain 954 + 180 = 1134.

How can a nearby convenient number help?

Worked example 12. Use multiplication with brackets to evaluate 97 × 25.

Answer: Write 97 as 100 − 3. Then 97 × 25 = (100 − 3) × 25 = 100 × 25 − 3 × 25. Evaluating the two products gives 2500 − 75 = 2425.

This method uses the distributive property with subtraction. The expression still describes 97 lots of 25, but the calculation uses 100 lots and removes three lots. Whether this feels quicker can depend on the particular numbers and the calculation method being compared.

The products 95 × 8, 104 × 15 and 49 × 50 can be approached in the same way, using 100 − 5, 100 + 4 and 50 − 1, respectively. Choose the rewritten form because it makes the products easier to handle, while preserving the original value.

How do we check that an expression answers a word problem?

A correct calculation must answer the quantity actually requested. Identify what each number measures, decide which amounts belong together, and check whether the final result counts money, packets, stories or something else. A number present in the question is not automatically needed in the expression.

How do repeated daily amounts combine?

The district market in Begur operates seven days a week. Rahim supplies 9 kg of mangoes each day and Shyam supplies 11 kg each day. Their combined weekly supply is 7 × (9 + 11), which gives 140 kg.

The bracket combines the two daily amounts before multiplying by the number of days. Alternatively, their weekly supplies can be counted separately and then added. Both methods must include each person's supply on all seven days.

How do regular expenses become yearly savings?

Binu earns ₹20,000 each month and spends ₹5,000 on rent, ₹5,000 on food and ₹2,000 on other expenses. For a year of twelve months, subtract the grouped monthly expenses from the monthly income and multiply the remainder by twelve.

Her yearly savings are 12 × (20000 − (5000 + 5000 + 2000)) = 96000 rupees. The same amount can be found as 12 × 20000 − 12 × (5000 + 5000 + 2000), subtracting yearly expenses from yearly income.

When should a number in the question be left unused?

Melvin reads one two-page story each day except Tuesdays and Saturdays. To count stories over eight weeks, use (7 − 2) × 8 or 7 × 8 − 2 × 8. These count five reading days each week and give 40 stories.

The two pages describe the length of each story. Multiplying by two would count pages instead of stories. This distinction is a check on the meaning of the expression, even when the arithmetic itself is straightforward.

Why does the final step sometimes need separate treatment?

A snail climbs 3 cm during the day and slips 2 cm at night on a 10 cm post; cm means centimetres. After seven complete day-and-night cycles it has gained 7 cm. On the eighth day it climbs the remaining 3 cm and reaches the treat.

The expression 7 × (3 − 2) + 3 = 10 describes reaching the top. Including another night-time slip before counting arrival would answer a different question. Check where the process ends before treating every day as an identical complete cycle.

How can number puzzles build skill with expressions?

An expression puzzle fixes the available numbers and asks you to combine them using permitted operations and brackets. The challenge is to change how the numbers are connected while respecting the conditions. Compare the values produced by different arrangements rather than assuming that the same numbers must give the same result.

What can three copies of one number produce?

Using three copies of 3, the expression (3 + 3) ÷ 3 gives 2. The brackets make the sum the quantity being divided. With the same three copies, 3 + 3 − 3 gives 3, while 3 × 3 + 3 gives 12.

These expressions use the same number three times, but their operations and grouping differ. In the last expression, the terms are 3 × 3 and 3. Evaluating the multiplication inside its term before addition produces the stated value.

What conditions should be checked?

One challenge is to use four copies of 4 to obtain values from 1 to 20. Another uses 1, 2, 3, 4 and 5 exactly once each to seek values between −10 and +10. A further challenge uses every number from 0 to 9 exactly once to make 100.

Check the conditions as well as the arithmetic: the required numbers must appear the specified number of times. Brackets must show the intended grouping, and the expression must evaluate to the desired value. A correct value alone does not show that the puzzle's conditions have been followed.

Glossary

• Arithmetic expression — A mathematical phrase formed using numbers and arithmetic operations, with a value obtained by evaluation.
• Value — The number obtained by carrying out the operations represented in an arithmetic expression.
• Operation — An action performed on numbers, such as addition, subtraction, multiplication or division.
• Sum — The result obtained when numbers or the evaluated terms of an expression are added.
• Product — The result obtained by multiplying numbers, as in counting several equal groups together.
• Equality sign — The symbol =, showing that the expressions or numbers on its two sides have equal values.
• Brackets — The symbols ( ), grouping an expression to be evaluated before the operations surrounding it.
• Term — A part separated by addition after subtraction is rewritten as addition of the inverse.
• Inverse — Here, the number with the opposite sign, added in place of subtracting the original number.
• Commutative property of addition — The property that swapping two terms, with their signs retained, does not change their sum.
• Associative property of addition — The property that changing how terms are grouped for addition does not change their sum.
• Distributive property — Multiplying a sum or difference by a number gives the sum or difference of the separate products.

Common errors and misconceptions

• Misconception: Different-looking expressions must have different values. Correct: Their operations can differ while their values agree. For example, 10 + 2, 15 − 3, 3 × 4 and 24 ÷ 2 all represent 12.
• Misconception: Evaluate 30 + 5 × 4 by adding 30 and 5 first. Correct: The terms are 30 and 5 × 4. Evaluate the product first, giving 30 + 20 = 50.
• Misconception: The terms of 83 − 14 are 83 and 14. Correct: Rewrite subtraction as addition of the inverse. The terms are 83 and −14, and the negative sign must be retained.
• Misconception: Every number in a product is a separate term. Correct: In 6 × 5 + 3, the whole product 6 × 5 is one term, and 3 is the other.
• Misconception: Removing brackets in 200 − (40 + 3) gives 200 − 40 + 3. Correct: Both parts of the sum must be subtracted, giving 200 − 40 − 3.
• Misconception: Subtracting a difference leaves both inner signs unchanged. Correct: In 500 − (250 − 100), removing brackets gives 500 − 250 + 100, because subtracting 250 removes too much.
• Misconception: 2 × (43 + 24) is the same as 2 × 43 + 24. Correct: Both bracketed terms must be multiplied, giving 2 × 43 + 2 × 24.
• Misconception: Melvin's two-page stories require multiplying the number of stories by two. Correct: That would count pages. The number of stories depends on reading days and weeks, not their page length.

Exam-style questions with model answers

Q1. Explain what an arithmetic expression and its value mean, using 13 + 2. [2 marks]

1. An arithmetic expression is a mathematical phrase involving numbers and operations; 13 + 2 represents adding 2 to 13.
2. Its value is the result of carrying out the operation. Here the value is 15, so 13 + 2 = 15.

Q2. Compare 1023 + 125 and 1022 + 128 without evaluating both totals. Give your reasoning. [3 marks]

1. Compare the starting numbers: 1023 is one more than 1022, so the first expression begins with an advantage of one.
2. Compare the amounts added: 128 is three more than 125, so the second expression gains three more than the first.
3. The extra three outweigh the initial one by two. Therefore, the second expression has the greater value: 1023 + 125 < 1022 + 128.

Q3. Identify the terms of 48 − 10 × 2 + 16 ÷ 2 and evaluate it, showing the treatment of signs and operations. [4 marks]

1. Rewrite the subtraction as addition of a negative term. The expression becomes 48 + (−10 × 2) + 16 ÷ 2.
2. The three terms are 48, −10 × 2 and 16 ÷ 2. Neither multiplication nor division separates a term.
3. Evaluate the multiplication and division within their terms: −10 × 2 = −20 and 16 ÷ 2 = 8.
4. Add the evaluated terms with their signs: 48 + (−20) + 8 = 36. Therefore, the value of the expression is 36.

Q4. Remove the brackets in 500 − (250 − 100), evaluate the expression, and explain why the final 100 is added. [3 marks]

1. The bracketed difference is 250 − 100 = 150, so the original expression asks for 150 to be subtracted from 500.
2. Subtracting 250 directly removes 100 more than required. Adding 100 back corrects this, giving the equivalent expression 500 − 250 + 100.
3. Evaluating gives 250 + 100 = 350. The final 100 is positive because the negative sign before the brackets reverses the signs of both inner terms.

Q5. Lhamo and Norbu each order one vegetable cutlet costing ₹43 and one rasgulla costing ₹24. Form two equivalent expressions for the total bill and explain the distributive property. Explain why 2 × 43 + 24 is unsuitable. [5 marks]

1. Each person orders the same pair of items. Therefore, the expression 43 + 24 describes the cost of one person's complete meal.
2. There are two complete meals. Multiplying the whole cost by two gives the expression 2 × (43 + 24) for their combined bill.
3. Alternatively, count the items separately. There are two vegetable cutlets and two rasgullas, so the combined bill is represented by 2 × 43 + 2 × 24.
4. The distributive property connects the two methods: 2 × (43 + 24) = 2 × 43 + 2 × 24. Both terms inside the brackets are multiplied.
5. The expression 2 × 43 + 24 counts only one rasgulla along with the two cutlets. It fails to include both people's complete meals.

Q6. Given 53 × 18 = 954, find 63 × 18 by using the known product. Explain each stage. [3 marks]

1. Since 63 is 53 + 10, rewrite the required product as (53 + 10) × 18. This represents ten more groups of 18.
2. Use the distributive property to obtain 53 × 18 + 10 × 18. The supplied product gives the first term as 954.
3. The extra ten groups total 180. Add this to the known product: 954 + 180 = 1134, so 63 × 18 = 1134.

Q7. Melvin reads one two-page story every day except Tuesdays and Saturdays. Taking seven days in each week, find how many stories he reads in eight weeks. Give two equivalent expressions and explain why multiplying by the page count is unnecessary. [5 marks]

1. Each week has seven days, but Tuesday and Saturday are excluded. Melvin therefore reads on 7 − 2 = 5 days each week.
2. He reads one story on each reading day. Over eight weeks, the expression (7 − 2) × 8 counts his stories, giving 40 stories.
3. An alternative is to count all days in eight weeks and subtract the excluded days. This gives the expression 7 × 8 − 2 × 8.
4. The distributive property shows that (7 − 2) × 8 and 7 × 8 − 2 × 8 have the same value and describe the same reading schedule.
5. The two-page detail describes the length of each story. Multiplying the story count by two would count pages, while the question asks for stories.

Key takeaways

• An arithmetic expression shows operations on numbers; its value is the number obtained after those operations are correctly evaluated.
• Compare expressions through their values, or reason about how changes in their numbers affect the complete expressions.
• Brackets group a quantity that must be evaluated before the surrounding operations are performed.
• Identify terms by replacing subtraction with addition of the inverse, keeping each negative sign attached to its term.
• The commutative and associative properties allow signed terms in a sum to be swapped and grouped without changing its value.
• When subtracting a bracketed sum or difference, removing the brackets changes the signs of its terms.
• The distributive property multiplies every term of a bracketed sum or difference by the number outside.
• Check what the question asks to count before forming an expression, especially when a problem contains extra descriptive numbers.

Test yourself

What does the equality sign tell you about two different-looking expressions?

It tells you that their values are equal, even if the numbers and operations are written differently.

What are the terms of 2 − 10 + 4 × 6?

The terms are 2, −10 and 4 × 6. Keep the negative sign with 10 and the whole product together.

Why does 30 + 5 × 4 give 50?

The product 5 × 4 is one term and evaluates to 20. Adding the other term, 30, gives 50.

What do the commutative and associative properties of addition allow?

The commutative property allows swapping terms; the associative property allows changing their grouping. Neither changes the sum.

Remove the brackets from 28 + (35 − 10).

The expression becomes 28 + 35 − 10. The grouped quantity is added, so its signed terms remain unchanged.

Why is 200 − (40 + 3) equal to 200 − 40 − 3?

Subtracting the whole sum means removing both 40 and 3. Subtracting them separately removes the same total amount.

Rewrite 14 × 10 − 6 × 10 using one pair of brackets.

It becomes (14 − 6) × 10. Removing six groups of ten from fourteen groups leaves eight groups.

A snail climbs 3 cm daily and slips 2 cm nightly on a 10 cm post. Why does it arrive on day eight?

After seven complete day-and-night cycles it has reached 7 cm. On day eight it climbs the remaining 3 cm and reaches the top before another slip.