BHA Grade 7 Math – Study Notebook

🔢 Integers, Order of Operations & Number Properties

PEMDAS / BODMAS
Parentheses → Exponents → Multiply/Divide → Add/Subtract
"Please Excuse My Dear Aunt Sally"

Calculate: $ 3 + 4 \times (2^2 - 1) \div 3 $

Step 1 — Parentheses first: $ 2^2 = 4 $, then $ 4 - 1 = 3 $

Step 2 — Multiply/Divide left→right: $ 4 \times 3 = 12 $, then $ 12 \div 3 = 4 $

Step 3 — Add: $ 3 + 4 = \mathbf{7} $

Q 01

Easy

Evaluate: $ 5 + 3 \times (8 - 2^2) \div 2 $
⚠️ Common mistake: forgetting exponent BEFORE subtract!

$ 5 + 3 \times (8 - 2^2) \div 2 $
Exponent first: $2^2 = 4$ → inside parentheses: $8-4=4$
Multiply: $3 \times 4 = 12$ → Divide: $12 \div 2 = 6$
Add: $5 + 6 = \mathbf{11}$ ✓

NEGATIVE × NEGATIVE = POSITIVE | NEGATIVE × POSITIVE = NEGATIVE
Same signs → (+) Different signs → (−)

Q 02

Easy

Simplify: $ (-4) \times (-3) + (-2) \times 5 $
⚠️ Tricky: do the two multiplications first, THEN add!

$(-4) \times (-3) = +12$ (same signs → positive)
$(-2) \times 5 = -10$ (different signs → negative)
$12 + (-10) = \mathbf{2}$ ✓

📊 Fractions, Decimals & Percentages

FRACTION DIVISION: "Keep, Change, Flip" (KCF)
$\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}$

⭐ Always simplify (reduce) your fraction at the END!

Q 03

Easy

Calculate: $ \dfrac{3}{4} \div \dfrac{9}{8} $

KCF: $\dfrac{3}{4} \div \dfrac{9}{8} = \dfrac{3}{4} \times \dfrac{8}{9} = \dfrac{24}{36}$
Simplify (÷12): $\dfrac{24}{36} = \mathbf{\dfrac{2}{3}}$ ✓

Q 04

Easy

A jacket costs $80. It is on sale for 35% off. What is the sale price?

Discount amount = $80 \times 0.35 = \$28$
Sale price = $80 - 28 = \mathbf{\$52}$ ✓
OR: Sale price = $80 \times 0.65 = \$52$

⚡ Simplifying Expressions & Solving Equations

LIKE TERMS: Same variable AND same power can be combined
$3x + 5x = 8x$ ✓ $3x + 5x^2 = $ can NOT combine ✗

Distributive Property

$ a(b + c) = ab + ac $

Simplify: $2(3x - 4) + 5x$

Step 1 — Distribute: $6x - 8 + 5x$

Step 2 — Combine like terms: $11x - 8$

Q 05

Easy

Simplify: $ 4(2x - 3) - 2(x + 1) $
⚠️ Watch the sign when distributing $-2$!

$4(2x-3) = 8x - 12$
$-2(x+1) = -2x - 2$
Combine: $(8x - 2x) + (-12 - 2) = \mathbf{6x - 14}$ ✓

SOLVING EQUATIONS — "INVERSE OPERATIONS"
Whatever you do to ONE side, do to the OTHER side too!
Goal: get the variable ALONE on one side.

Q 06

Easy

Solve for $x$: $ 3x - 7 = 2x + 5 $

Subtract $2x$ from both sides: $x - 7 = 5$
Add 7 to both sides: $x = \mathbf{12}$ ✓
Check: $3(12)-7 = 29$ and $2(12)+5 = 29$ ✓

Q 07

Easy

Solve for $n$: $ \dfrac{2n + 4}{3} = 6 $
⚠️ Multiply BOTH sides by 3 first — don't divide 6 by 3 alone!

Multiply both sides by 3: $2n + 4 = 18$
Subtract 4: $2n = 14$
Divide by 2: $n = \mathbf{7}$ ✓

📉 Substitution & Algebraic Thinking

If $x = -2$, be CAREFUL: $x^2 = (-2)^2 = 4$ NOT $-4$!

Q 08

Easy

If $a = -3$ and $b = 2$, find the value of: $ a^2 - 3ab + b^2 $

$a^2 = (-3)^2 = 9$
$-3ab = -3(-3)(2) = +18$
$b^2 = 2^2 = 4$
Total: $9 + 18 + 4 = \mathbf{31}$ ✓

⚖️ Ratios, Proportions & Scale

PROPORTION — CROSS MULTIPLY
If $\dfrac{a}{b} = \dfrac{c}{d}$, then $a \times d = b \times c$

Unit Rate Formula

Unit Rate $= \dfrac{\text{Total amount}}{\text{Number of units}}$

Q 09

Easy

A map has a scale of $1 : 50{,}000$. Two cities are $6$ cm apart on the map. What is the actual distance in km?

Real distance $= 6 \times 50{,}000 = 300{,}000$ cm
Convert: $300{,}000 \div 100{,}000 = \mathbf{3}$ km ✓
100,000 cm = 1 km

Q 10

Easy

The ratio of boys to girls in a class is $3 : 5$. There are $24$ boys. How many girls are in the class?
⚠️ Don't find total students — find girls only!

Boys part = 3, Girls part = 5
1 part = $24 \div 3 = 8$
Girls = $8 \times 5 = \mathbf{40}$ ✓

📐 2D Shapes: Area & Perimeter

Key Formulas — MEMORIZE THESE!

Circle: $A = \pi r^2$ | $C = 2\pi r$ Triangle: $A = \frac{1}{2}bh$ Trapezoid: $A = \frac{1}{2}(a+b)h$

r vs d: radius = HALF of diameter | $r = d \div 2$
ALWAYS check: did the problem give you r or d?

Q 11

Easy

A circle has a diameter of $10$ cm. What is its area? (Use $\pi \approx 3.14$)
⚠️ Most common mistake: using diameter instead of radius!

Diameter = 10, so radius $r = 5$ cm
$A = \pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = \mathbf{78.5 \text{ cm}^2}$ ✓

Q 12

Easy

Find the area of a trapezoid with parallel sides of $6$ cm and $10$ cm, and a height of $4$ cm.

$A = \frac{1}{2}(a + b) \times h = \frac{1}{2}(6 + 10) \times 4$
$= \frac{1}{2} \times 16 \times 4 = \mathbf{32 \text{ cm}^2}$ ✓

📦 3D Shapes: Surface Area & Volume

Volume Formulas

Rectangular Prism: $V = l \times w \times h$ Cylinder: $V = \pi r^2 h$ Triangular Prism: $V = \frac{1}{2}bh \times l$

Q 13

Easy

A cylinder has a radius of $3$ cm and a height of $5$ cm. Find its volume. (Use $\pi \approx 3.14$)

$V = \pi r^2 h = 3.14 \times 3^2 \times 5 = 3.14 \times 9 \times 5$
$= 3.14 \times 45 = \mathbf{141.3 \text{ cm}^3}$ ✓

📏 Angles & Coordinate Geometry

ANGLE RULES:
Supplementary = adds to 180° | Complementary = adds to 90°
Vertically opposite angles are EQUAL | Triangle angles sum = 180°

Q 14

Easy

Two angles are supplementary. One angle is $ (3x + 10)° $ and the other is $ (x + 30)° $. Find $x$.
⚠️ Supplementary = SUM is 180°, not each angle equals 180°!

Supplementary → sum = 180°
$(3x+10) + (x+30) = 180$
$4x + 40 = 180$ → $4x = 140$ → $x = \mathbf{35}$ ✓

Q 15

Easy

Point A is at $(2, -3)$. It is reflected over the x-axis. What are the coordinates of A'?
💡 Reflect over x-axis → y-coordinate changes sign. Reflect over y-axis → x changes sign.

Reflection over x-axis: keep x, negate y
$(2, -3) \rightarrow \mathbf{(2, 3)}$ ✓
Over y-axis would give $(-2, -3)$ — don't mix up!

📊 Mean, Median, Mode & Range

3 M's + R:
Mean = sum ÷ count | Median = middle value (sorted!) |
Mode = most frequent | Range = max − min

⚠️ ALWAYS sort the data set before finding median!

Q 16

Easy

Data set: $\{7, 12, 5, 12, 9, 3, 12, 6\}$
Which measure of central tendency best represents this data?
⚠️ Think: what makes mode special here?

Mean = $(7+12+5+12+9+3+12+6) \div 8 = 66 \div 8 = 8.25$
Sorted: $3,5,6,7,9,12,12,12$ → Median = $(7+9)/2 = 8$
Mode = 12 (appears 3 times) ✓
When one value repeats most, mode best represents the trend!

Q 17

Easy

The mean of 5 numbers is $14$. Four of the numbers are $10, 16, 12, 18$. What is the fifth number?

Total sum needed = $14 \times 5 = 70$
Sum of 4 known = $10+16+12+18 = 56$
Fifth number = $70 - 56 = \mathbf{14}$ ✓

🎲 Probability

Basic Probability

$P(\text{event}) = \dfrac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$ $0 \leq P \leq 1$

COMPLEMENT RULE: $P(\text{not A}) = 1 - P(A)$
"Everything that's NOT happening" = 1 minus what IS happening

Q 18

Easy

A bag contains 4 red, 3 blue, and 5 green marbles. What is the probability of NOT picking a green marble?

Total marbles = $4+3+5 = 12$
$P(\text{green}) = \frac{5}{12}$
$P(\text{not green}) = 1 - \frac{5}{12} = \mathbf{\frac{7}{12}}$ ✓
Or directly: $\frac{4+3}{12} = \frac{7}{12}$

📈 Patterns & Linear Relationships

LINEAR PATTERN: Find the "common difference" (constant change)
$n$-th term formula: $T_n = a + (n-1)d$ | $a$=first term, $d$=common difference

Q 19

Easy

A sequence starts: $4, 7, 10, 13, \ldots$
What is the 15th term?

First term $a = 4$, common difference $d = 3$
$T_{15} = 4 + (15-1) \times 3 = 4 + 42 = \mathbf{46}$ ✓
Quick check: $T_1=4, T_2=7, T_3=10$ ✓

Q 20

Easy

The graph of $y = 2x - 3$ passes through which of these points?
💡 Substitute each x-value and check if you get the right y-value!

Test each:
A: $y = 2(0)-3 = -3 \neq 3$ ✗
B: $y = 2(2)-3 = 1$ ✓ ← YES!
C: $y = 2(-1)-3 = -5$ ← also correct! But let's verify B: $(2,1)$: $2(2)-3 = 1$ ✓
D: $y = 2(3)-3 = 3 \neq 4$ ✗
Both B and C work — but answer is $\mathbf{(-1, -5)}$ ... wait, check C: $y=2(-1)-3=-2-3=-5$ ✓
Answer C: $(-1,-5)$ ✓

🎉 Correct!

📐 Grade 7 Mathematics
Study Notebook

🎓 Quiz Complete!

🎉 Correct!

📐 Grade 7 MathematicsStudy Notebook

🎓 Quiz Complete!

📐 Grade 7 Mathematics
Study Notebook