Equations and Inequalities: practice solutions, Grade 6

Rate this post

Practice solutions Grade 6 : Equations and Inequalities — Zyro the alien explorer of Planète Maths

Written solutions to the chapter problems. Check each step, then correct yourself.

1 Is it a solution? ★★★

$8 + 6 = 14$, not 15. Not a solution.
$3 \times 7 = 21$. Solution.
$14 - 5 = 9$. Solution.
$12 \div 4 = 3$. Solution.

2 Adding and subtracting ★★★

Subtract 7: $x = 19 - 7 = 12$.
Subtract 2.5: $y = 6 - 2.5 = 3.5$.
Subtract 9: $m = 14 - 9 = 5$.

Check: $12 + 7 = 19$, $3.5 + 2.5 = 6$, $5 + 9 = 14$.

3 Multiplying and dividing ★★★

Add 8: $n = 19$.
Divide by 6: $p = 9$.
Multiply by 5: $k = 35$.
Divide by 0.5: $w = 12$.

Check: $19 - 8 = 11$; $6 \times 9 = 54$; $35 \div 5 = 7$; $0.5 \times 12 = 6$.

4 Choose the inverse operation ★★★

Subtract 12: $x = 18$.
Add 3: $x = 13$.
Divide by 8: $x = 5$.
Multiply by 6: $x = 12$.

5 From words to equations ★★★

$n - 7 = 20$, so $n = 27$.
$3n = 36$, so $n = 12$.
$\dfrac{n}{4} = 9$, so $n = 36$.
$n + 12 = 31$, so $n = 19$.

6 Reading a number line ★★★

The circle is open at 5 and the ray goes right: $x > 5$.
6 and 9 are solutions. 4 is too small, and 5 is not included because the circle is open.

7 Test the numbers ★★★

Only numbers strictly greater than 6: 9 and 12.
Numbers less than or equal to 6: 3 and 6.

Notice that 6 is a solution of (b) but not of (a).

8 Stickers ★★★

Let $s$ be the number of stickers at first. Equation: $s + 17 = 45$. Subtract 17: $s = 28$.

Check: $28 + 17 = 45$. Maya had 28 stickers.

9 Concert tickets ★★★

Equation: $6t = 45$. Divide both sides by 6: $t = 7.5$.

Check: $6 \times 7.5 = 45$. One ticket costs $7.50.

10 A weekend hike ★★★

Equation: $d + 3.5 = 9.2$. Subtract 3.5: $d = 5.7$.

Check: $5.7 + 3.5 = 9.2$. Leo hiked 5.7 miles on Saturday (about 9.2 km).

11 Fractions and decimals ★★★

Subtract $\dfrac{1}{4}$: $x = \dfrac{3}{4} - \dfrac{1}{4} = \dfrac{2}{4} = \dfrac{1}{2}$.
Add $\dfrac{2}{3}$: $y = \dfrac{1}{3} + \dfrac{2}{3} = 1$.
Divide by 0.4: $x = 2.8 \div 0.4 = 7$.

12 Find the mistake ★★★

Sam added 9 to 21, but the equation already adds 9 to $x$. He must use the inverse operation, subtraction.

Correct work: $x = 21 - 9 = 12$. Check: $12 + 9 = 21$.

13 Solve the inequalities ★★★

Subtract 5: $x > 7$.
Add 3: $y \leq 12$.
Divide by 4: $n < 7$.
Multiply by 3: $m \geq 18$.

14 Graph an inequality ★★★

The symbol $\geq$ includes 3, so the circle is closed. The ray goes to the right.

15 The elevator ★★★

Total weight must be at most 1,200: $640 + w \leq 1200$. Subtract 640: $w \leq 560$.

They can add at most 560 pounds (about 254 kg).

16 Counting solutions ★★★

Whole numbers start at 0. The whole numbers less than 5 are 0, 1, 2, 3 and 4: that is five solutions, so Tom is wrong. (Numbers like 4.9 are solutions too, but they are not whole numbers.)

17 The garden rectangle ★★★

Area = length times width: $7l = 91$. Divide by 7: $l = 13$ cm.
Perimeter $= 2 \times (13 + 7) = 2 \times 20 = 40$ cm.

Check: $7 \times 13 = 91$.

18 Sharing a bill ★★★

$9x = 135$.
Divide by 9: $x = 15$. Each pays $15.
With 8 friends: $8y = 135$, so $y = 16.875$, about $16.88. That is about $1.88 more each ($16.875 - 15 = 1.875).

19 Tall enough to ride ★★★

$44.5 + g \geq 48$. Subtract 44.5: $g \geq 3.5$.
No: $3 < 3.5$, so she would be 47.5 inches, still too short.

20 Two conditions at once ★★★

First: subtract 4, so $x < 7$. Second: divide by 3, so $x > 4$.

Whole numbers greater than 4 and less than 7: 5 and 6. Check 6: $6 + 4 = 10 < 11$ and $18 > 12$.

21 Saving for a bike ★★★

$64 + s \geq 150$, so $s \geq 86$.
$9w \geq 86$, so $w \geq \dfrac{86}{9} \approx 9.56$.
Weeks are counted in whole numbers: 9 weeks give $81, which is not enough; 10 weeks give $90, which is enough.

22 Error in an inequality ★★★

Jo subtracted 5, but 5 multiplies $x$. The inverse operation is division: $x \leq 40 \div 5 = 8$.

Inside: $x = 6$ gives $30 \leq 40$, true. Outside: $x = 9$ gives $45 \leq 40$, false. With Jo’s answer, $x = 30$ would be allowed but $5 \times 30 = 150$ is far too big.