Question paper, Set 4

Integers are $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$: no fractions and no decimals.

So the integers are $32, -25, 11, 0$

Answer: $32, -25, 11, 0$

Leave out $-8$ and $3$ themselves and do not miss $0$.

$-7, -6, -5, -4, -3, -2, -1, 0, 1, 2$

Counting them: $7 + 1 + 2 = 10$ integers

Answer: $-7, -6, -5, -4, -3, -2, -1, 0, 1, 2$; there are $10$ integers

Both are negative: add them and keep the $-$ sign.

$-29 - 98 = -(29 + 98)$

$= -127$

Answer: $-127$

This is $-39 + 61$.

The signs are different: subtract the smaller from the bigger and keep the sign of the bigger.

$-39 + 61 = +(61 - 39)$

$= 22$

Answer: $22$

Going up adds.

$-4 + 12 = +(12 - 4) = 8$

Answer: Floor $8$

Both are negative: add them and keep the $-$ sign.

$-755 - 2497 = -(755 + 2497)$

$= -3252$

Answer: $-3252$

Profit $= +959$, loss $= -7967$

$959 - 7967 = -(7967 - 959) = -7008$

The answer is $-7008$: a loss of ₹7008.

Answer: $-7008$, that is, a loss of ₹7008

Subtract column by column from the units, borrowing when the top digit is smaller.

(i) Units: $7 - 2 = 5$; Tens: $13 - 6 = 7$ (borrow 1); Hundreds: $11 - 9 = 2$ (borrow 1); Thousands: $4$; so $5237 - 962 = 4275$

(ii) Units: $13 - 8 = 5$ (borrow 1); Tens: $9 - 3 = 6$ (borrow 1); Hundreds: $11 - 9 = 2$ (borrow 1); Thousands: $17 - 8 = 9$ (borrow 1); Ten thousands: $8$; so $98203 - 8938 = 89265$

(iii) Units: $11 - 7 = 4$ (borrow 1); Tens: $3 - 2 = 1$; Hundreds: $12 - 9 = 3$ (borrow 1); Thousands: $9 - 3 = 6$ (borrow 1); Ten thousands: $4$; so $50241 - 3927 = 46314$

Answer: (i) $4275$, (ii) $89265$, (iii) $46314$

Different signs and the bigger number $637$ is negative, so $-637 + 294 = -(637 - 294)$

Count down: $637 \to 300$ is $337$, and $300 \to 294$ is $6$

$637 - 294 = 337 + 6 = 343$

So $-637 + 294 = -343$

Answer: $-343$

Different signs and the bigger number $1274$ is negative, so $-1274 + 741 = -(1274 - 741)$

Count down: $1274 \to 800$ is $474$, and $800 \to 741$ is $59$

$1274 - 741 = 474 + 59 = 533$

So $-1274 + 741 = -533$

Answer: $-533$

Positive terms: $18 + 967 = 985$

Negative terms: $812 + 382 = 1194$

$18 - 812 + 967 - 382 = 985 - 1194$

$= -(1194 - 985) = -209$

Answer: $-209$

(i) Positive terms: $831 + 534 = 1365$

Negative terms: $295 + 406 = 701$

$831 + 534 - 295 - 406 = 1365 - 701$

$= 664$

(ii) Positive terms: $2215 + 5453 = 7668$

Negative term: $3849$

$2215 - 3849 + 5453 = 7668 - 3849$

$= 3819$

Answer: (i) $664$, (ii) $3819$

$+ \times - = -$

$19 \times -36 = -(19 \times 36) = -684$

Answer: $-684$

(i) $+ \times - = -$, $57 \times -9 = -513$

(ii) $- \times - = +$, $-53 \times -7 = 371$

Answer: (i) $-513$, (ii) $371$

(i) $+ \times - = -$; $71 \times 3 = 213$, $71 \times 30 = 2130$, $213 + 2130 = 2343$

So $71 \times -33 = -2343$

(ii) $- \times + = -$; $681 \times 6 = 4086$, $681 \times 70 = 47670$, $4086 + 47670 = 51756$

So $-76 \times 681 = -51756$

(iii) $- \times - = +$; $905 \times 6 = 5430$, $905 \times 60 = 54300$, $5430 + 54300 = 59730$

So $-905 \times -66 = 59730$

Answer: (i) $-2343$, (ii) $-51756$, (iii) $59730$

$\displaystyle \frac{+}{-} = -$

So $\displaystyle \frac{6112}{-8}$ is negative.

Answer: Negative

$\displaystyle \frac{-}{+} = -$

$8 \div 5 = 1$ r $3$; $35 \div 5 = 7$; $3 \div 5 = 0$ r $3$; $35 \div 5 = 7$

$8535 \div 5 = 1707$, so $\displaystyle \frac{-8535}{5} = -1707$

Answer: $-1707$

(i) $\displaystyle \frac{+}{-} = -$

$6 \div 6 = 1$; $5 \div 6 = 0$ r $5$; $55 \div 6 = 9$ r $1$; $13 \div 6 = 2$ r $1$; $18 \div 6 = 3$

$65538 \div 6 = 10923$, so $\displaystyle \frac{65538}{-6} = -10923$

(ii) $\displaystyle \frac{-}{-} = +$

$6 \div 5 = 1$ r $1$; $12 \div 5 = 2$ r $2$; $20 \div 5 = 4$; $3 \div 5 = 0$ r $3$; $36 \div 5 = 7$ r $1$; $10 \div 5 = 2$

$620360 \div 5 = 124072$, so $\displaystyle \frac{-620360}{-5} = 124072$

Answer: (i) $-10923$, (ii) $124072$

(i) Addition, different signs: $-11 + 5 = -(11 - 5) = -6$

(ii) Multiplication: $- \times + = -$, so $-11 \times 5 = -55$

Answer: (i) $-6$, (ii) $-55$

Addition uses the addition rule; multiplication and division use the sign rule.

(i) $-5 - 36 = -(5 + 36) = -41$

(ii) $- \times - = +$, $-5 \times -36 = 180$

(iii) $-29 + 9 = -(29 - 9) = -20$

(iv) $- \times + = -$, $-29 \times 9 = -261$

(v) $\displaystyle \frac{+}{-} = -$; $9 \div 6 = 1$ r $3$; $30 \div 6 = 5$; $5 \div 6 = 0$ r $5$; $55 \div 6 = 9$ r $1$; $18 \div 6 = 3$; so $\displaystyle \frac{90558}{-6} = -15093$

Answer: (i) $-41$, (ii) $180$, (iii) $-20$, (iv) $-261$, (v) $-15093$