Mathematical Operations with New Symbols

What you will learn

You will read a symbol definition, rewrite an expression correctly and then calculate. There are two kinds of task here. In a renamed-operator task, a printed sign stands for a familiar arithmetic operation. In a defined two-input operation, a new sign stands for a formula involving its left and right inputs. Do not mix these two meanings.

You need the earlier arithmetic lesson's grouping, signed calculation and equal-precedence rules. Coding mastery is not required. This lesson adds careful interpretation of new symbols; it does not replace ordinary arithmetic with a new universal BODMAS rule.

1 The legend comes before the calculation

A legend is a small symbol-to-meaning table. If @ means +, read an original @ as addition. Rewrite the whole expression using ordinary signs before doing any arithmetic. A familiar printed sign can also be renamed, so its ordinary appearance alone does not decide its meaning.

For these substitution questions, replace each original token exactly once, simultaneously. A token is one printed sign in the original expression. Numbers and parentheses stay in place. Once translated, the new signs are ordinary arithmetic signs; do not translate them again.

Worked example

Here @ means +, # means ×, and $ means −. Replace all symbols once, then use ordinary order of operations.

8 @ 3 # 4 $ 5

Answer: 15

8+3×4−5=8+12−5=15.

Watch the mistake: Pure left-to-right arithmetic would give 39; it ignores multiplication priority after translation.

The first line of a good solution is the translated expression. It separates two possible mistakes: reading the legend incorrectly and calculating the correct translation incorrectly. Keeping these stages visible makes a wrong answer easier to repair.

2 Arithmetic priority applies after translation

Use the ordinary convention on the completed ordinary expression:

1. Evaluate the innermost parentheses or other explicit grouping first
2. Evaluate powers when present
3. Do multiplication and division at the same priority, from left to right
4. Do addition and subtraction at the same priority, from left to right

The letters in BODMAS do not mean “always divide before multiplying” or “always add before subtracting”. A rule such as “read all operators left to right regardless of their type” would be a different rule and would need explicit instruction. It is not used for the translated arithmetic here.

3 Swapping familiar signs: one simultaneous pass

Suppose + is given the meaning ÷ and ÷ is given the meaning −. An original + becomes ordinary ÷ once. It does not then become −. You are interpreting the original printed expression, not repeatedly running a text replacement until no symbol remains.

Worked example

In the original expression, + means ÷, ÷ means −, − means ×, and × means +. Replace each original symbol once, simultaneously; then use ordinary precedence.

24 + 6 − 2 × 5 ÷ 1

Pause: translate only

Original tokens: 24 | + | 6 | − | 2 | × | 5 | ÷ | 1

Ordinary row: 24 | □ | 6 | □ | 2 | □ | 5 | □ | 1

Fill only the four operator blanks using the original tokens. Do not calculate until the completed ordinary expression has been checked. Each original sign changes once; a newly inserted sign is not sent through the legend again.

Answer: 12

Translate to 24÷6×2+5−1; 4×2+5−1=8+5−1=12.

Watch the mistake: Never feed a newly substituted sign through the legend again. Division and multiplication are evaluated left to right.

The four completed sign blanks are ÷, ×, +, −. Now 24÷6×2 is evaluated left to right: 4×2=8. The remaining +5−1 gives 12. Multiplying 6×2 first would change the arithmetic grouping and give the wrong result.

4 Brackets are part of the information

A parenthesis groups a complete input to a later operation. Keep it in exactly the same place when translating symbols. Without brackets, addition and subtraction share a priority and are evaluated left to right; brackets can deliberately change that order.

Worked example

@ means − and # means +. Replace once, then use ordinary precedence and preserve all brackets.

18 @ 7 # 3

18 @ (7 # 3)

Answer: 14; 8

18−7+3=11+3=14, while 18−(7+3)=18−10=8.

Watch the mistake: Addition is not automatically before subtraction; brackets deliberately change the grouping.

In the first expression the subtraction leaves 11, then 3 is added. In the second, the whole bracket has value 10 and is subtracted from 18. These are two different expressions, not competing methods for the same expression. Never silently insert a bracket to suit a mnemonic.

5 A new operation defined by a formula

In a ★ b = 2a+b, the symbol ★ does not simply mean multiplication or addition. It names the whole instruction “double the left input, then add the right input”. For a single use, identify a and b, substitute them into the definition and calculate.

The definition also does not guarantee that inputs can be exchanged or brackets moved. For example, 3★4 uses left input 3 and right input 4, while 4★3 uses the opposite order. With nested operations, evaluate the displayed inner bracket first and treat its result as one outer input.

Worked example

For any numbers a and b, define a ★ b = 2a+b. Evaluate only the fully parenthesised expressions given.

(3 ★ 4) ★ 5

3 ★ (4 ★ 5)

Pause: fill the inner/outer input table for (3 ★ 4) ★ 5

• Inner operation: left input 3; right input 4; output □
• Outer operation: left input □; right input 5; output □

Complete the inner result and copy it into the outer left-input slot. Keep the original right input 5. Then apply the definition again. Solve the separately bracketed second expression only after this trace is clear.

Answer: 25; 19

3★4=10, then 10★5=25. Separately, 4★5=13, then 3★13=19.

Watch the mistake: ★ has no assumed ordinary-arithmetic precedence or associativity. Operand order also matters: 3★4=10 but 4★3=11.

The first table completes as inner output 10, outer left input 10, outer right input 5, outer output 25. In the second expression, the inner result 13 is instead the outer right input, so the output is 19. This is why moving the brackets is not allowed merely because the same three numbers appear.

A bare chain such as a★b★c has no defined grouping here. Our questions show the brackets rather than silently assigning an arithmetic precedence to ★. Also, if a future defined formula includes division, first check that its denominator is not zero; an undefined expression cannot be repaired by selecting a convenient numeric option.

6 A two-stage checking routine

For renamed operators:

• Copy the original tokens and the complete legend
• Replace each original operator once, keeping every number and bracket
• Read back the ordinary expression against the legend
• Apply ordinary precedence, including left-to-right ties
• Check sign and size where useful

For a formula-defined operation:

• Copy the exact definition and identify left/right inputs
• Work inside the displayed brackets first
• Substitute the result into the correct outer input position
• Apply the definition again without changing input order or grouping

Independent practice

Try all four before the explained key. These original learning questions are free and untimed, with +1 for correct and 0 for incorrect or unattempted, no negative marking and no pass cutoff. Their local settings are not a claim about RRB exam marking. Each question includes the relevant symbol meanings and evaluation convention.

1. Here @ means + and # means ×. Replace each original symbol once, then multiply before adding. Find 7 @ 4 # 3.

A. 33 B. 14 C. 19 D. 84

2. In the original expression, + means ×, × means −, − means ÷, and ÷ means +. Replace all original signs once, simultaneously. Then do × and ÷ left to right before + and − left to right. Evaluate 24 − 4 + 2 × 3 ÷ 1.

A. 1 B. 26 C. 8 D. 10

3. Here @ means −, # means +, and $ means ×. Replace original signs once, keep the brackets, then use ordinary precedence: brackets first, multiplication before subtraction. Evaluate 30 @ (8 # 3) $ 2.

A. 8 B. 38 C. 28 D. 16

4. Define a ★ b = 3a−b. Preserve the left/right input order and the displayed brackets. Find (5 ★ 2) ★ 4.

A. 13 B. 35 C. −1 D. 9

Explained key

1. C — 19

Translate first: 7+4×3. Multiplication gives 12, then 7+12=19. A evaluates the translated expression purely left to right: (7+4)×3=33. B treats both signs as addition, giving 7+4+3=14. D treats both as multiplication, giving 7×4×3=84. Neither B nor D follows the two different legend entries.

2. D — 10

Once-only translation gives 24÷4×2−3+1. Divide and multiply left to right: 6×2=12. Then subtract and add left to right: 12−3+1=10. A wrongly multiplies 4×2 first, obtaining 24÷8−3+1=1. B evaluates the original printed signs without translation: 24−4+6=26. C adds 3+1 before subtracting, as if the expression were 12−(3+1)=8.

3. A — 8

The translated expression is 30−(8+3)×2. The bracket is 11, so 30−11×2=30−22=8. B incorrectly groups the first subtraction: (30−11)×2=38. C ignores the given bracket: 30−8+3×2=28. D applies the factor 2 only to 3 inside a changed bracket: 30−(8+3×2)=16. The original parentheses must survive translation unchanged.

4. B — 35

The inner operation is 5★2=3×5−2=13. The outer left input is now 13, so 13★4=3×13−4=35. A regroups as 5★(2★4): 2★4=2, then 5★2=13. C reverses the inner inputs: (2★5)★4=1★4=−1. D omits the outer multiplication by 3 and does only 13−4=9.

Source and scope

RRB CEN 09/2025, §14.1, printed p.28, names alphabetical and number series, coding and decoding, and mathematical operations. The topic list is illustrative, not exhaustive. OpenStax Prealgebra 2e, §2.1 states the grouping and equal-priority arithmetic convention used here. All explanations, examples and questions here are original teaching material. The selected subtypes and question counts are learning choices, not official topic weightage. These are not previous-year questions or a full reasoning course.

Analogy

Think of a translator followed by a calculator. The translator reads each original sign once and writes its instructed arithmetic meaning while leaving numbers and brackets in place. Only the completed ordinary expression goes to the calculator. A formula-defined operation is different: it is a small two-input machine with labelled left and right slots, so the inner machine’s output must enter the correct outer slot.

Quick reference

• Read the complete symbol legend or formula
• Replace each original operator token exactly once, simultaneously
• Keep numbers, operand order and every bracket
• Calculate only after ordinary-sign translation
• Ordinary order: grouping, powers, ×/÷ left to right, +/− left to right
• Never send an inserted ordinary sign back through the legend
• A custom operation uses its exact formula, with labelled left/right inputs
• Fully parenthesise custom chains; do not assume regrouping or input exchange is allowed
• Reject zero denominators; undefined is not a numerical answer