Exercises — Operations on decimals — all four operations

L1.1 Align the points, pad the shorter one: $6.30+12.85$ Now both share denominator $100$: $\frac{630}{100}+\frac{1285}{100}=\frac{1915}{100}$. Add the whole numbers $630+1285=1915$; because the points were aligned, the answer's point drops straight down (see the column figure above) ⇒ $\boxed{19.15}$.

L1.2 $0.4$ has $1$ place, $0.007$ has $3$ places ⇒ product has $1+3=4$ places. Strip the points (read them as plain digits): $4\times 7=28$. Give it $4$ places: $28\to 0.0028$. $0.4\times 0.007=\boxed{0.0028}$

L1.3 The divisor $0.3$ has $1$ decimal place ⇒ multiply both by $10^1=10$: $9÷0.3=\frac{9\times 10}{0.3\times 10}=\frac{90}{3}=\boxed{30}$

L2.1 Pad all to $2$ places so every number shares denominator $100$: $23.60+4.08+0.70$ Add whole numerators: $2360+408+70=2838$, i.e. $\frac{2838}{100}$. Point straight down ⇒ $\boxed{28.38}$.

L2.2 Write $15$ as $15.000$ (pad to match $3$ places), so both are over $1000$: $15.000-2.375=\frac{15000-2375}{1000}$ The borrowing across the point: the thousandths column reads $0-5$, which is short, so we borrow $1$ from the hundredths column (worth $10$ thousandths). This borrowing cascades leftward — $0-7,0-3,4-2\dots$ — exactly as it would for the whole numbers $15000-2375$. Because we padded, the padded zeros are real zeros in those columns, so borrowing across the decimal boundary is no different from borrowing across any other column. Result: $15000-2375=12625$. $\frac{12625}{1000}=\boxed{12.625}$

L2.3 Strip the points: $25\times 24=600$. Places: $2.5$ has $1$, $0.24$ has $2$ ⇒ total $3$. Give $600$ three places: $600\to 0.600$. $2.5\times 0.24=\boxed{0.6}$

L2.4 Divisor $0.8$ has $1$ place ⇒ multiply both by $10$: $6.72÷0.8=\frac{67.2}{8}$ Long-divide keeping the point above its spot: $67.2÷8=8.4$. $\boxed{8.4}$ Verify (inverse): $8.4\times 0.8=6.72$ ✓.

L3.1 $0.6=\frac{6}{10}$, a fraction less than $1$. Multiplying by something less than $1$ shrinks the number, so $3.4\times 0.6<3.4$. Exact: strip the points $34\times 6=204$; places $1+1=2$ ⇒ $2.04$. $3.4\times 0.6=\boxed{2.04}(2.04<3.4✓)$

L3.2 This is a division in disguise: blank $=3÷0.12$. Divisor $0.12$ has $2$ places ⇒ multiply both by $100$: $\frac{3\times 100}{0.12\times 100}=\frac{300}{12}=25.$ Blank $=\boxed{25}$. Check: $0.12\times 25=3$ ✓ (see Fractions to Decimals conversion — $0.12=\frac{12}{100}$).

L3.3 $0.75\times 100$: multiplying by $10^2$ shifts the point $2$ places right ⇒ $\boxed{75}$. $0.75÷0.25$: divisor $0.25$ has $2$ places ⇒ $\frac{75}{25}=\boxed{3}$. Multiplying/dividing by a power of ten shifts the point; dividing by an ordinary number ($0.25$) does not — after clearing the divisor's point it is just plain division.

L4.1 Brackets first (align points): $2.50+1.30=3.80$. Then multiply: strip the points $380\times 4=1520$; places $2+1=3$ ⇒ $1.520=1.52$. $\boxed{1.52}$

L4.2 Division before subtraction. Divisor $0.9$ has $1$ place ⇒ $\frac{72}{9}=8$. So $7.2÷0.9=8$. Then $8-1.5=6.5$. $\boxed{6.5}$

L4.3 Cost of $8$ pencils: $0.35\times 8$. Strip the points $35\times 8=280$; places $2+0=2$ ⇒ $2.80$. Change: $5.00-2.80=2.20$. $\text{change}=\boxed{\$2.20}$ Verify: $2.80+2.20=5.00$ ✓.

L4.4 $0.6\times 0.6$: strip $6\times 6=36$; places $1+1=2$ ⇒ $0.36$. Then $0.36-0.25=0.11$ (align to $2$ places). $\boxed{0.11}$

L4.5 The smaller number is on top, so the answer is negative. Swap, subtract, attach the minus: $5.75-2.30=3.45$, therefore $2.3-5.75=-3.45=\boxed{-3.45}$

L5.1 Division: divisor $1.2$ has $1$ place ⇒ $\frac{14.4}{12}=1.2$. Why the shift preserves the quotient: we multiplied top and bottom by the same $10$, and $\frac{1.44\times 10}{1.2\times 10}=\frac{1.44}{1.2}$ — a fraction is unchanged when both parts are scaled equally. Verify: $1.2\times 1.2=1.44$ ✓. Multiplication: $0.3\times 2.5$: strip $3\times 25=75$; places $1+1=2$ ⇒ $0.75$. Verify: $0.75÷2.5=0.3$ ✓. Add: $1.20+0.75=1.95$. $\boxed{1.95}$

L5.2 Servings $=1.6÷0.25$. Divisor $0.25$ has $2$ places ⇒ $\frac{160}{25}=6.4$. You can make only whole servings ⇒ $\lfloor 6.4\rfloor =6$ full servings. Milk used: $6\times 0.25=1.5$ L. Left over: $1.6-1.5=0.1$ L. $\boxed{6\text{ servings, }0.1\text{ L left}}$

L5.3 $0.008=\frac{8}{10^3}=\frac{8}{1000}$. Multiply by $0.5$: strip $8\times 5=40$; places $3+1=4$ ⇒ $0.0040=0.004$. As a fraction: $\frac{8}{1000}\times \frac{5}{10}=\frac{40}{10^4}=\frac{4}{1000}=0.004$. Scientific notation: $0.004=4\times 10^{-3}$. $\boxed{0.004=4\times 10^{-3}}$

L5.4 Dividing by $0$ asks "what number times $0$ gives $4.5$?" — but anything times $0$ is $0$, never $4.5$, so no such number exists: the operation is undefined. The shift rule tries to make the divisor whole by multiplying by $10^k$, but $0.0\times 10^k=0$ for every $k$ — the divisor stays $0$, so the rule cannot help. Division by zero is forbidden, decimals or not.

+ / −: align the points (pad with zeros); if the top is smaller, answer is negative. ×: multiply plain (stripped) numbers, then place a point with as many decimals as the two factors' places added; apply the sign rule. ÷: shift both numbers by the same $10^k$ to make the divisor whole (this scales top and bottom equally, so the quotient is unchanged), then long-divide — but never divide by $0$.