Why it happens: Every subtraction can be turned into an addition of the opposite number. Written as a sum of terms, an expression becomes much easier to swap and group.
Q2.
Example 8: Charu’s scores in three rounds of a quiz are 7p – 3q, 8p – 4q, and 6p – 2q, where p is the score for a correct answer and q the penalty for an incorrect answer. What do each of the expressions mean?
Answer
Round
Expression
What it says
1
7p – 3q
7 correct answers and 3 wrong answers
2
8p – 4q
8 correct answers and 4 wrong answers
3
6p – 2q
6 correct answers and 2 wrong answers
Why it happens: Each correct answer earns p marks, so 7 of them earn 7p. Each wrong answer costs q marks, so 3 of them cost 3q — a subtraction.
Q3.
If the score for a correct answer is 4 (p = 4) and the penalty for a wrong answer is 1 (q = 1), find Charu’s score in the first round.
Try This: Another set is 10p – 4q, 6p – 1q and 7p – 2q. Make up a third set of your own — there are many.
Q8.
Can we say who scored more? Can you explain why? (Charu: 21p – 9q, Krishita: 23p – 7q)
Answer
Krishita scored more.
Krishita has 23p, Charu has 21p → Krishita gains 2p more Krishita loses 7q, Charu loses 9q → Krishita loses 2q less
Why it happens: Krishita wins on both counts — more correct answers and fewer wrong ones. Since p and q are positive numbers, her total must be larger, whatever the marking scheme is.
Q9.
How much more has Krishita scored than Charu? Simplify this expression further: 23p – 7q – (21p – 9q)
Why it happens: 4 times a sum is the sum of 4 times each part. After that, 4y and –y are like terms, and 4y – y is 3y.
Q11.
Example 10: Are the expressions 5u and 5 + u equal to each other?
Answer
No. They describe two different operations.
5u means 5 times the number u 5 + u means 5 more than the number u
Why it happens: Two expressions are equal only if they give the same value for every value of the letter. Here they agree only at u = 1.25, and differ everywhere else — so they are not equal expressions.
Q12.
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. (u = 11, 2, 8 and 5)
Are the expressions 10y – 3 and 10(y – 3) equal? Let us compare the values that these expressions take for different values of y (y = 2, 0, 7, 10). After filling in the two diagrams, do you think the two expressions are equal?
Answer
10y – 3 means ‘3 less than 10 times y’, while 10(y – 3) means ‘10 times (3 less than y)’.
y
10y – 3
10 (y – 3)
2
17
–10
0
–3
–30
7
67
40
10
97
70
y = 2: 10 × 2 – 3 = 17, but 10 × (2 – 3) = 10 × (–1) = –10 y = 0: 0 – 3 = –3, but 10 × (–3) = –30 y = 7: 70 – 3 = 67, but 10 × 4 = 40 y = 10: 100 – 3 = 97, but 10 × 7 = 70
No, the two expressions are not equal.
Why it happens: Opening the bracket shows the gap clearly: 10(y – 3) = 10y – 30, which is 27 less than 10y – 3 for every y.
Q14.
Example 11: What is the sum of the numbers in the picture (unknown values are denoted by letter-numbers)? (four 3s on top, r and s, r and s, four 3s at the bottom)
Why it happens: The picture is symmetric top-to-bottom, so the third method works. All three expressions look different but simplify to the same thing — that is what makes them equal expressions.