Apply the same relationship to complete a number analogy - CogAT Grades 3 Plus

1. If $8 : 24 = 12 : ?$, then what is the missing number?
   Answer: 36 The relationship is $8 \times 3 = 24$, so $12 \times 3 = 36$. Choice A (30) represents adding 18 to both sides incorrectly. Choice C (40) comes from incorrectly thinking the pattern is adding 32. Choice D (48) results from multiplying 12 by 4 instead of 3.
2. Look at the coordinate plane below. The slope from line segment $RS$ is applied again from $T$ to find $U$. What is the $x$-coordinate of $U$?
   Answer: $7$ R(2,1) to S(4,3) moves +2x, +2y. Apply to T(5,4): +2x →7.
3. In the analogy $5 \times 3 : 30 = 7 \times 4 : ?$, what is the missing value?
   Answer: 56 $5 \times 3 = 15$ and $15 \times 2 = 30$, so $7 \times 4 = 28$ and $28 \times 2 = 56$. Choice A (28) stops at $7 \times 4$ without applying the doubling relationship. Choice C (84) results from multiplying 28 by 3 instead of 2. Choice D (168) comes from multiplying 28 by 6.
4. What number makes this analogy true? $28 : 7 = 56 : ?$
   Answer: 14 The relationship is $28 \div 7 = 4$, so $56 \div ? = 4$, which means $? = 14$. Choice A (12) comes from incorrectly using $56 \div 4.67$. Choice C (16) results from thinking $56 \div 3.5 = 16$. Choice D (18) comes from adding 11 to 7 and applying to 56 incorrectly.
5. Complete this analogy: $18 : 2 = 54 : ?$
   Answer: 6 $18 \div 2 = 9$, so $54 \div ? = 9$, which gives $? = 6$. Choice A (4) comes from incorrectly thinking $54 \div 13.5$. Choice C (8) results from using $54 \div 6.75$. Choice D (9) confuses the quotient with the divisor.
6. Based on the bar graph, the height relationship from Bar 1 to Bar 2 repeats from Bar 3 to which height?
   Answer: $18$ cm Bar1 12 cm, Bar2 9 cm: subtract 3. Bar3 21 cm minus 3 = 18 cm.
7. In the analogy $15 : 3 = 35 : ?$, what number completes the relationship?
   Answer: 7 The relationship is $15 \div 3 = 5$, so $35 \div 5 = 7$. Choice A (5) confuses the quotient with the answer. Choice C (12) comes from incorrectly subtracting 3 from 15 and applying that to 35. Choice D (15) results from thinking the second number stays constant.
8. What number completes this relationship? $\frac{36}{9} : 2 = \frac{48}{6} : ?$
   Answer: 4 $\frac{36}{9} = 4$ and $4 \div 2 = 2$, so $\frac{48}{6} = 8$ and $8 \div 2 = 4$. Choice A (1) comes from incorrectly dividing 8 by 8. Choice B (2) confuses the division factor with the answer. Choice D (8) stops at $\frac{48}{6}$ without applying the division by 2.
9. Use the diagram below. The first arrow shows how $4$ becomes $1$. Apply the same rule to $12$.
   Answer: $3$ $4 \div 4 =1$, so divide by $4$ again: $12\div4=3$. Distractors divide by other numbers.
10. Refer to the figure below. The arrows show the change from Box 1 to Box 2. Apply the same change from Box 3 to find the number that should go in Box 4.
    Answer: $7$ Box1 $14$ to Box2 $21$ means add $7$. Box3 holds $0$, so Box4 should be $0+7 =7$. Choices B, C, D add larger amounts.
11. In the proportion $\frac{4}{12} = \frac{7}{?}$, expressed as an analogy $4 : 12 = 7 : ?$, what is the missing number?
    Answer: 21 $\frac{4}{12} = \frac{1}{3}$, so $\frac{7}{?} = \frac{1}{3}$, which means $? = 21$. Choice A (18) comes from incorrectly multiplying 7 by 2.57. Choice C (24) results from adding 12 to both terms. Choice D (28) comes from multiplying 7 by 4.
12. Complete this number analogy: $45 : 5 = 72 : ?$
    Answer: 8 $45 \div 5 = 9$, so we need $72 \div ? = 9$, which gives us $? = 8$. Choice A (6) comes from incorrectly using $72 \div 12$. Choice C (9) confuses the quotient with the missing divisor. Choice D (12) results from incorrectly using $72 \div 6$.
13. If $6^2 : 9 = 8^2 : ?$, then the missing value is:
    Answer: 16 $6^2 = 36$ and $36 \div 4 = 9$, so $8^2 = 64$ and $64 \div 4 = 16$. Choice A (12) comes from incorrectly finding $8^2 \div 6$. Choice C (18) results from adding 9 to the difference between 8 and 6. Choice D (24) comes from multiplying 8 by 3.

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