NUMBER

$$6+2\times4-1 = 6+8-1 = 13$$

Example results: $(6+2)\times4-1 = 31$; $6+2\times(4-1)=12$; $(6+2)\times(4-1)=24$; $6+(2\times4-1)=13$. (Working shown for each.)

Bracketing the addition together with a following multiplication tends to increase the result (since addition happens before a larger multiplier is applied), while bracketing to delay multiplication tends to decrease it. Of the options tested, $(6+2)\times4-1=31$ gives the largest value.

Step 1 (powers): $2^2=4$. Step 2 (multiplication/division, left to right): $3\times4=12$ and $8\div4=2$. Step 3 (addition/subtraction, left to right): $15-12+2=5$.

Without an agreed order, the same expression could be evaluated in different sequences, giving different answers depending on who solves it. A universal convention ensures every mathematician gets the same, unambiguous result for the same expression.

$$\text{Total} = (3\times4.50 + 2\times3.20)\times1.10 - 2$$ Subtotal $= 13.50+6.40=19.90$. After service charge: $19.90\times1.10=21.89$. After discount: $21.89-2=19.89$.

Yes, \$21.89 matches the subtotal-plus-service-charge stage of the calculation, before the \$2 voucher is subtracted. The correct final amount the customer should pay is \$19.89.

Brackets: $24+36=60$. Multiply: $60\times3=180$. Subtract: $180-15=165$. Divide: $165\div5=33$. Power: $2^3=8$. Add: $33+8=41$.

New order: powers first ($2^3=8$), then multiply/divide ($36\times3=108$, $15\div5=3$), then add/subtract left to right: $24+108-3+8=137$. This differs greatly from 41 because without brackets, $36$ is multiplied by 3 alone (not the combined $60$), and $15$ is divided by 5 alone (not subtracted from a running total first) — brackets fundamentally change WHICH numbers get combined by which operations.

No brackets: $2+12-1=13$. $(2+3)\times4-1=20-1=19$. $2+3\times(4-1)=2+9=11$. $2+3\times4-1$ with brackets around $(3\times4-1)$: this doesn't change anything since multiplication already happens first, giving $2+11=13$ (same as no brackets).

3 genuinely different results: 13, 19, 11. The brackets around $(3\times4-1)$ didn't change anything because multiplication ALREADY happens before subtraction by the standard order of operations rule — adding brackets around an operation that would happen first anyway has no effect.

Student's left-to-right: $20-4=16$, $16\times3=48$, $48+2=50$, $50^2=2500$. Correct method: $2^2=4$ first, then $4\times3=12$, then $20-12+4=12$.

The methods diverge immediately at the SECOND step — left-to-right treats $-4$ then $\times3$ sequentially without prioritizing multiplication, and (critically) applies squaring to the WRONG number entirely (50 instead of just 2). A universal, agreed-upon order of operations ensures every person (and every calculator) evaluates any given expression identically, avoiding ambiguity.

$$45+38\times3.5-20 = 45+133-20=158$$

Customer's version: $(45+38)\times3.5-20 = 83\times3.5-20=290.5-20=270.5$. Discrepancy: $270.5-158=\$112.50$ (a significant overestimate). In a real billing context, this kind of misreading could cause a customer to be alarmed by an incorrectly high expected charge, or a business to genuinely OVERCHARGE a customer if the formula itself were programmed incorrectly — showing why precise, unambiguous mathematical formulas matter in real invoicing systems.

Brackets: $24+36=60$. Multiply: $60\times3=180$. Subtract: $180-15=165$. Divide: $165\div5=33$. Power: $2^3=8$. Add: $33+8=41$.

New order: powers first ($2^3=8$), then multiply/divide ($36\times3=108$, $15\div5=3$), then add/subtract left to right: $24+108-3+8=137$. This differs greatly from 41 because without brackets, $36$ is multiplied by 3 alone (not the combined $60$), and $15$ is divided by 5 alone (not subtracted from a running total first) — brackets fundamentally change WHICH numbers get combined by which operations.

No brackets: $2+12-1=13$. $(2+3)\times4-1=20-1=19$. $2+3\times(4-1)=2+9=11$. $2+3\times4-1$ with brackets around $(3\times4-1)$: this doesn't change anything since multiplication already happens first, giving $2+11=13$ (same as no brackets).

3 genuinely different results: 13, 19, 11. The brackets around $(3\times4-1)$ didn't change anything because multiplication ALREADY happens before subtraction by the standard order of operations rule — adding brackets around an operation that would happen first anyway has no effect.

Student's left-to-right: $20-4=16$, $16\times3=48$, $48+2=50$, $50^2=2500$. Correct method: $2^2=4$ first, then $4\times3=12$, then $20-12+4=12$.

The methods diverge immediately at the SECOND step — left-to-right treats $-4$ then $\times3$ sequentially without prioritizing multiplication, and (critically) applies squaring to the WRONG number entirely (50 instead of just 2). A universal, agreed-upon order of operations ensures every person (and every calculator) evaluates any given expression identically, avoiding ambiguity.

$$45+38\times3.5-20 = 45+133-20=158$$

Customer's version: $(45+38)\times3.5-20 = 83\times3.5-20=290.5-20=270.5$. Discrepancy: $270.5-158=\$112.50$ (a significant overestimate). In a real billing context, this kind of misreading could cause a customer to be alarmed by an incorrectly high expected charge, or a business to genuinely OVERCHARGE a customer if the formula itself were programmed incorrectly — showing why precise, unambiguous mathematical formulas matter in real invoicing systems.